methods of analysis for earthquake- resistant structures

6

Methods of Analysisfor Earthquake-

Resistant Structures

6.1 Introduction

Structural Analysis Procedures • Models of StructuresLoads and Boundary Conditions • Notation

6.2 Equilibrium

Node Equilibrium • Equilibrium of Basic System of Element Forces • Lower Bound Theorem of Plastic Analysis

6.3 Geometric Compatibility

Displacement–Deformation Relationship under Large Displacements • Linear Approximation of Displacement–Deformation Relation • Compatibility Relationship for Elements with Moment Releases

6.4 Equilibrium in Deformed Configuration

6.5 Principle of Virtual Work

Virtual Work Principles for the Structure • Virtual Work Principles for an Element • Upper Bound Theorem of Plastic Analysis • Example of Plastic Analysis

6.6 Force–Deformation Relationships

Section Response • Material Models

6.7 Frame Elements

Basic Relationships • Concentrated Plasticity Elements •Distributed Inelasticity Elements

6.8 Solution of Equilibrium Equations

Newton–Raphson Iteration • Load Incrementation • Load Factor Control during Incrementation • Load Factor Control during Iteration • Structure State Determination • State Determination of Elements with Force Formulation • Nonlinear Solution of Section State Determination

6.9

Nonlinear Geometry and P-

D

Geometric Stiffness

Resisting Forces and Element Tangent Stiffness Matrix •Geometric Stiffness Matrix •

P-

D

Geometric Stiffness

6.10 Examples of Nonlinear Static Analysis

6.11 Dynamic Analysis

Free Vibration • Modal Analysis for Linear Response •Earthquake Excitation • Numerical Integration of Equations of Motion for Linear Response • Numerical Integration of Equations of Motion for Nonlinear Response

6.12 Applications of Linear and Nonlinear Dynamic Analysis

6.13 Conclusions

6.14 Future Challenges

Filip C. Filippou

Gregory L. Fenves

© 2004 by CRC Press LLC

6

-2

Earthquake Engineering: From Engineering Seismology to Performance-Based Engineering

6.1 Introduction

This chapter presents structural analysis methods for designing earthquake-resistant structures. The focusis on structural models consisting of frame elements for modeling beam, column and brace members,which is the common type of modeling for buildings and bridges in current earthquake engineeringpractice. Structural analysis software often used in engineering design incorporates one or more of themethods of analysis presented in this chapter. The chapter discusses sources of nonlinear material andgeometric behavior. It covers plastic analysis methods for collapse load and plastic deformation deter-mination under the assumption of elastic-perfectly plastic material response. It briefly describes nonlinearhysteretic material models and uses these to derive the hysteretic response of sections under the interactionof biaxial moment and axial force. This chapter further discusses concentrated inelasticity frame elements,and compares two approaches for the derivation of the force–deformation relation of distributed inelas-ticity frame elements. The effect of nonlinear geometry is presented in the general form of the corotationalformulation for frame elements under large displacements. Consistent approximations are introducedto arrive at simplified nonlinear geometry methods that suffice in many design situations, in particular,the so-called

P

-

D

geometric stiffness. The chapter concludes with a discussion of linear and nonlineardynamic response under ground excitation. The key features of the analysis methods in this chapter areillustrated with examples of static and dynamic nonlinear response of components and structures. Thechapter concludes with a few important observations and a discussion of future challenges for improvingstructural analysis procedures for earthquake-resistant design.

6.1.1 Structural Analysis Procedures

The analysis of a structural system to determine the deformations and forces induced by applied loadsor ground excitation is an essential step in the design of a structure to resist earthquakes. A structuralanalysis procedure requires: (i) a model of the structure, (ii) a representation of the earthquake groundmotion or the effects of the ground motion and (iii) a method of analysis for forming and solving thegoverning equations. There is a range of methods from a plastic analysis to a sophisticated nonlinear,dynamic analysis of a detailed structural model that can be used, depending on the purpose of the analysisin the design process. This chapter presents structural analysis procedures for earthquake-resistant design.The focus is on methods for structural models consisting of frame elements for modeling beam, columnand brace members, which is the common type of modeling for buildings and bridges. Structural wallsare often modeled with beam elements at the centerline and rigid joint offsets, even though this doesnot properly account for the uplift effect that may be important for this type of lateral load resistingsystem.

An important decision in a structural analysis is to assume whether the relationship between forcesand displacements is linear or nonlinear. Linear analysis for static and dynamic loads has been used instructural design for decades. Nonlinear analysis methods are widely used, because emerging perfor-mance-based guidelines require representation of nonlinear behavior. There are two major sources ofnonlinear behavior. The first is a nonlinear relationship between force and deformation resulting frommaterial behavior such as ductile yielding, stiffness and strength degradation or brittle fracture. Thesecond type of nonlinear behavior is caused by the inclusion of large displacements in the compatibilityand equilibrium relationships. This chapter presents the nonlinear methods of analysis for both types ofbehavior. Linear methods are a special case.

An earthquake analysis generally includes gravity loads and a representation of the ground motion atthe site of the structure. Earthquake ground motion induces the mass in a structure to accelerate, andthe resulting response history can be computed by dynamic analysis methods. In many design proceduresit is common to perform a dynamic analysis with a response spectrum representation of the groundmotion expected at the site (Chopra, 2001). For response history analysis, several analyses with differentground motion histories of the earthquake hazard are generally required. (See Chapter 5 for moreinformation about the definition of earthquake ground motion in a structural analysis.)


Methods of Analysis for Earthquake-Resistant Structures

6

-3

For many design procedures, however, it is common to use equivalent static loads that represent theeffects of the earthquake on the structure. Traditional design procedures use a static linear analysis witha response modification coefficient to represent the effects of ductile, nonlinear behavior. In contrast,newer design procedures utilize a nonlinear static (pushover) analysis to determine the force–displace-ment relationship for the structure, and the inelastic deformation of its members.

After a structural model and earthquake loading are defined, an analysis method is needed to computethe response. The governing equations are formed using equilibrium, compatibility and force–deforma-tion relationships for the elements and the structure, and are expressed in terms of unknown displace-ments (or degrees of freedom, referred to as DOFs in this chapter). To elucidate the theory and providea compact mathematical representation, the fundamental relationships are expressed using matrix alge-bra. Since the governing equations may have a large number of degrees of freedom, they must be solvednumerically using a computer-based analysis method. Nearly all structural analyses for earthquake-resistant design are performed using software that incorporates one or more of the analysis methodspresented in this chapter. Modern software generally includes graphical features for visualizing the forcesand deformations computed from an analysis. Before using any new structural analysis software, theengineer should conduct an independent verification to ensure that the software provides correct solutions.

The structural analysis procedures used in earthquake-resistant design are summarized in Table 6.1.Recent guidelines for seismic rehabilitation of buildings pioneered the requirements for dynamic andnonlinear analysis procedures, particularly FEMA 356 (FEMA, 2000a) and the predecessor FEMA 273(FEMA, 1997). The ATC-40 guidelines for reinforced concrete buildings (ATC, 1996b) emphasize theuse of a nonlinear static (pushover) analysis procedure to define the displacement capacity for buildings.The classification of analysis procedures in Table 6.1 is generally applicable to design regulations for newbuildings, such as in the 2000 NEHRP recommended provisions (FEMA, 2001) and recent guidelinesfor steel moment frame buildings (FEMA, 2000b, c) and for bridges (ATC, 1996a). These provisions andguidelines are required for the selection of the analysis procedure depending on the seismic designcategory, performance level, structural characteristics (e.g., regularity or complexity), response charac-teristics (e.g., the fundamental vibration period and participation of higher vibration modes), amountof data available for developing a model and confidence limits (in a statistical sense) for performanceevaluation. Design provisions for structures with seismic isolation systems and supplemental energydissipation generally require a dynamic analysis procedure.

The analysis procedures in Table 6.1 are in order of increasingly rigorous representation of structuralbehavior, but also increasing requirements for modeling and complexity of the analysis. As described inSection 6.2, plastic analysis only requires the equilibrium relationships, and is useful for capacity designprocedures (Paulay and Priestley, 1992). For a given load distribution and flexural strength of members,plastic analysis gives the collapse load and the location of plastic hinges in members. The linear staticprocedure has been a traditional structural analysis method for earthquake-resistant design (UBC, 1997),

TABLE 6.1

Structural Analysis Procedures for Earthquake-Resistant Design

CategoryAnalysis Procedure

Force–Deformation Relationship Displacements

Earthquake Load

Analysis Method

Equilibrium Plastic Analysis Procedure

Rigid-plastic Small Equivalent lateral load

Equilibrium analysis

Linear Linear Static Procedure

Linear Small Equivalent lateral load

Linear static analysis

Linear Dynamic Procedure I

Linear Small Response spectrum

Response spectrum analysis

Linear Dynamic Procedure II

Linear Small Ground motion history

Linear response history analysis

Nonlinear Nonlinear Static Procedure

Nonlinear Small or large Equivalent lateral load

Nonlinear static analysis

Nonlinear Dynamic Procedure

Nonlinear Small or large Ground motion history

Nonlinear response history analysis


6

-4


but it does not represent the nonlinear behavior or the dynamic response of a structure caused by anearthquake ground motion. The simplest dynamic analysis method is based on a linear model of thestructure, which permits use of vibration properties (frequencies and mode shapes) and simplificationof the solution with a modal representation of the dynamic response. An estimate of the maximumstructural response can be obtained with response spectrum analysis, or the maximum can be computedby response history analysis with specific earthquake ground motion records. Linear dynamic analysismethods are covered in depth by Chopra (2001).

Increasingly, engineers are using, and design guidelines are requiring, nonlinear analysis in the designprocess, because a severe earthquake ground motion is expected to deform a structure into the inelasticrange. Nonlinear analysis methods can provide the relationship between a lateral load representing the effectof the earthquake ground motion and the displacements of the structure and deformations of the members.The results are often presented as a pushover or capacity curve for the structure. More detailed responsehistory of a structure (sometimes called the seismic demand) can be computed by nonlinear dynamicanalysis methods, particularly the cyclic response, degradation and damage measures for the members.

For the earthquake analysis of many types of structures it is reasonable to assume that the foundationand soil are rigid compared to the structure itself and that the supports of the structure move in phaseduring an earthquake ground motion. Soil–structure interaction, as described in Chapter 4, modifies theinput motion to a structure because of wave propagation and energy dissipation in the soil, however,this phenomenon is not discussed in this chapter. For two-dimensional analysis the ground motion isspecified in the horizontal and vertical directions; the two horizontal and the vertical ground motioncomponents are specified for three-dimensional analysis. The assumption of uniform ground motionmay not be valid for long-span bridges because of wave passage effects, differential site response andincoherence of the ground motion.

6.1.2 Models of Structures

A structural analysis is performed on a model of the structure — not on the real structure — so theanalysis can be no more accurate than the assumptions in the model. The model must represent thedistribution and possible time variation of stiffness, strength, deformation capacity and mass of thestructure with accuracy sufficient for the purpose of the analysis in the design process.

All structures are three dimensional, but it is important to decide whether to use a three-dimensionalmodel or simpler two-dimensional models. The analysis methods are the same whether the model istwo-dimensional or three-dimensional. Generally, two-dimensional models are acceptable for buildingswith regular configuration and minimal torsion; otherwise, a three-dimensional model is necessary witha representation of the floor diaphragms as rigid or flexible components. Analysis of bridges is generallybased on three-dimensional models, although nonlinear analysis is typically used for two-dimensionalmodels of bridge piers (ATC, 1996a, Priestley et al., 1996).

A structural model of a frame consists of an assembly of frame elements connected at nodal points(or nodes) in a global coordinate system, as illustrated in Figure 6.1. The geometry of the structuralmodel is described by the position of the nodes in a global coordinate system, denoted by X, Y and Z.In the graphic representations of the structural model, nodes have a small black square (see Figure 6.1).

Two nodes define a frame element,which may be either straight or curved.This chapter is limited tostraight elements because a curved element can always be approximated by several straight elements atthe expense of increased modeling effort and computational cost. The element geometry is establishedin a local coordinate system

x

,

y

,

z

(see Figure 6.1). As will be shown later in this chapter, the force–defor-mation relationship for the element is obtained from the integration of functions of

x

along the elementaxis between the nodes. These functions represent the section forces (such as shear, bending and axialforces), the corresponding section deformations and the relationship between section forces and defor-mations.

The element response can be completely described by the relation between the force vector

p

and thedisplacement vector

u

. For three-dimensional (3d) elements, the force vector has 12 components: at each



6

-5

node there are three forces in the local

x

,

y

,

z

coordinate system and three moments about the axes of thelocal coordinate system. In the two-dimensional (2d) case there are two forces and one moment at each node,providing six components of the force vector. The displacement vector is defined in an analogous mannerand includes translations in the direction of the local axes and rotations about the local axes at each node.

Before concluding this section on structural modeling with frame elements, it is worth noting that formany structural systems the joints connecting members may be substantial in size and have modes ofdeformation that affect the system behavior. In such cases the model needs to account for at least thefinite size of the joint region. The topic of joint deformation is more advanced than the scope of thischapter but models for joints can be included in the analysis methods presented herein.

Finite element methods with continuum elements (Bathe, 1995) can provide a more refined distribu-tion of stress and strain in solid, plate and shell models of structural components such as walls, dia-phragms and joints, but with the exception of walls, this level of refinement is generally not warrantedfor earthquake-resistant design. This chapter does not discuss the details of finite element methods forcontinuum elements, although most of the solution methods are applicable for models that includecontinuum finite elements as well as frame elements.

6.1.3 Loads and Boundary Conditions

Loads are specified forces applied to elements or nodes. Gravity loads may be applied to elements orconsidered as nodal loads depending on the gravity load path. The vector of nodal loads for a structureis denoted by

P

, with six components of force at each node for 3d problems and three components for2d problems. In contrast with nodal loads, element loads are included in the element force–deformationrelationship as distributed loads

w

(

x

) defined in the local coordinate system for the element.Each node undergoes translations and rotations that can be combined into a displacement vector of

three translations and three rotations at each node in the 3d case. The displacements of all nodes arecollected into a single displacement vector

U

for the entire model in which each component is a degreeof freedom. We separate the set of all global DOFs into two subsets: the DOFs with unknown displacementvalues and the DOFs with specified displacement value. Each DOF in the model must be included inone of the two sets. The unknown displacements are called the free DOFs and are denoted by

U

f

. Thesecond set of displacements corresponds to the restrained DOFs and is denoted by

U

d

. The restrainedDOFs are generally assigned a value of zero to indicate a fixed displacement, but nonzero supportdisplacement problems can be considered. The selection of restrained displacements at the supports isan important step in the structural modeling, and the supports of a model are commonly identified withthe symbols shown in Figure 6.2 for typical 2d cases. The arrows in Figure 6.2 indicate the restrainedDOFs, and thus the corresponding support reactions of each support type.

FIGURE 6.1

Local and global reference systems and notation.

X

Y

global coordinate system

local coordinate system

x

y

i

jb

element b

p,u

P,U


6

-6


Since the displacements are partitioned into two sets, so is the nodal force vector,

P

. The nodal forcesat the free DOFs of the model are specified as nodal loads, and are denoted by

P

f

. For earthquake analysisthis would normally include only the gravity loads with all other nodal loads equal to zero. The forcesat the restrained DOFs are the support reactions and are denoted by

P

d

. These can be evaluated oncethe equations for the free DOFs are solved.

6.1.4 Notation

A consistent notation assists in elucidating the fundamental structural analysis concepts. In general,uppercase symbols are matrices or vectors representing the structural system (Table 6.2a), whereaslowercase symbols represent quantities associated with individual elements (Table 6.2b). Vectors andmatrices are written in boldface.

6.2 Equilibrium

All structural analysis methods in Table 6.1 require satisfaction of equilibrium. This section presents thefundamental equilibrium relationships for nodes and elements in the model.

6.2.1 Node Equilibrium

With imaginary cuts around each node we separate

nn

nodal-free bodies and

ne

element-free bodies, asshown in Figure 6.3, in which

nn

is the number of nodes and

ne

is the number of elements in the model.Three force equilibrium equations and three moment equilibrium equations must be satisfied for each

FIGURE 6.2

Typical support symbols for the 2d case.

TABLE 6.2

Notation for Structural Analysis

Quantity Symbol

(a) Symbols for Structural (Global) System

Global coordinate system for structure X,Y,ZDisplacements of structure at DOFs

U

Applied loads to structure DOFs

P

Resisting forces for structure at DOFs

P

r

Structure equilibrium matrix for DOFs

B

Structure compatibility matrix for DOFs

A

Structural stiffness, mass and damping matrices

K, M, C

(b) Symbols for Elements

Local coordinate system for element x,y,zBasic element deformations

v

Element nodal displacements in local coordinate system and global coordinate system

_u, u

Basic element forces

q

Element nodal forces in local and global coordinate system

_p, p

el

Basic element flexibility and stiffness matrices

f, k



6

-7

node-free body for the 3d case and two force equilibrium and one moment equilibrium equations for anode in the 2d case. Using the 2d case for simplicity and examining the nodal equilibrium equations, itis apparent that the equilibrium equations involve the summation of element forces at a node when allelement forces are expressed in the global coordinate system. An element is identified with a superscriptin parentheses and the correspondence between element DOF and global DOF can be supplied by anarray, known as an

id

-array with the number of entries equal to the number of element DOFs. With therelationship between element and global DOF for each element, the force vector for an element

el

canbe mapped to the contribution of the element force vector to the global resisting force vector, representedsymbolically as , in which has nonzero terms corresponding to the forces in the element,as represented by the

id

-array. Using this relationship the nodal equations of static equilibrium are writtenas:

(6.1)

Equation 6.1 states that the applied forces

P

consisting of forces at free DOFs

P

f

and forces at restrained

DOFs

P

d

are in equilibrium with the resisting forces

P

r

, which are the sum of the element

contributions

p

(

el

)

. The mapping of element DOFs to global DOFs followed by summation of the element

contributions, as denoted by the symbol , indicates assembly of all element contributions, which is

known as the direct assembly procedure.Equation 6.1 assumes that the loading is applied so slowly that the resulting accelerations at the free

DOFs can be neglected. If this is not the case, the equations of static equilibrium are extended usingNewton’s second law stating that

P

–

P

r

=

M

(6.2)

in which is the acceleration with respect to a fixed frame of reference. Equation 6.2 is also knownas the equation of motion for a structural model. It is worthwhile to state explicitly the dependence ofthe resisting forces on the displacement and displacement rate (velocity) vector.

which allows for velocity-dependent resistance (viscous damping).

FIGURE 6.3

Node and element free body equilibrium.

P

p(c)

a

b c

d

p(b)

p P( ),

( )elidelÆ r Pr,

( )idel

P P 0P

PP 0 P P pA- =

ÊËÁ

ˆ¯

- = = =Ârf

dr r r,or with

id

el

elel

el( ) ( )

Ael

Ut

Ut

P P U Ur r= ( )ˆ , ˙


6

-8


6.2.2 Equilibrium of Basic System of Element Forces

Turning attention to the free body for a 3d frame element, there are 12 unknown forces, with six at theimaginary cuts at each end of the element but only six independent equilibrium equations. Figures 6.4aand b show the end forces in the global and local coordinate systems for a 2d frame element, with thequantities in the local system denoted by a bar. A rotation matrix can be used to transform vectorsbetween the two coordinate systems.

Because the element forces need to satisfy the equilibrium equations they are not independent. Weselect a subset of element forces and express the remainder in terms of the subset to assure that theequilibrium of the free body is satisfied. The independent element forces are called the basic forces

,

q

.In the 2d case there are three basic forces, and in this chapter we select the axial force and the two endmoments, as shown in Figure 6.4c. The three equilibrium equations for the element-free body are usedto express the remaining element forces in terms of the basic forces.

The equilibrium equations can be satisfied in the undeformed configuration, if the displacements aresmall relative to the dimensions of the structure. However, if the displacements are large, the equilibriumneeds to be satisfied in the deformed configuration. The latter leads to nonlinear geometric effects, whichare presented after the geometric compatibility relationships are defined for large displacements. Equi-librium of an element-free body in the undeformed configuration is shown in Figure 6.4c and stated asfollows:

FIGURE 6.4

Element forces in global, local and basic reference systems.

pi

pj

(a)

(b)

(c)

p−i

p−j

q1

q2

q3

L

L

q2 + q3

L

q2 + q3

q1

q1

3qwy

wxq2 q3

q1 + wxL

2

wy L

L−

q2 + q3

2

wy L

L+

q2 + q3

(d)



6

-9

(6.3)

If element loads are included, 6.3 is modified to include the effect of the element loads,

where is the vector of element forces due to the loads on the element. Figure 6.4d shows these forcesfor the case of uniform transverse and longitudinal distributed loads. The matrix

b

represents theequilibrium transformation matrix from the basic system to the complete system of element forces inlocal coordinates. We noted earlier that the transformation of the element end forces from the local tothe global coordinate system involves a rotation transformation of the end forces at each end, expressedby . Combining 6.3 with the rotation to the global coordinate system gives the element forces inthe global system expressed in terms of the basic forces and element loads:

(6.4)

The statement of element equilibrium can be extended to other cases, such as for rigid-end offsets to representfinite joint size. The element forces at the ends of the deformable portion of the element are denoted by ,and the equilibrium of the rigid-end offsets is shown in Figure 6.5. The equilibrium relationship is ,

FIGURE 6.5

Force transformation (equilibrium) for rigid-end offsets.

L

p2

p3

p4

p5p6

i

j

cxj

cyj

cyi

cxi

p~1

p1

p2

p3 cyi

cxi

X

Yp1

cxj

cyj

p4

p5

p6

p~5

p~4p~6

p~4

p~5

p~6

p~1

p~2

p~3

p~ 2 p~3

p q

pq q

p q

p q

pq q

p q

p

p

p

p

p

p

p

1 1

22 3

3 2

4 1

52 3

6 3

1

2

3

4

5

6

1 0 0

01 1

0 1 0

1 0 0

01 1

0 0 1

= -

=+

=

=

= -+

=

=

Ê

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜˜˜˜˜˜˜

=

-

- -

È

Î

ÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙

L

L

L L

L L

or,

˙˙

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜ =

q

q

q

bq1

2

3

p bq p= + w

pw

p b p= r

p b bq b p= +r r w

pp b p= j

˜


6

-10


and noting that , the element forces become . In a compact notationthe element equilibrium in the undeformed configuration can be stated as

(6.5)

in which the equilibrium matrix transforms the basic forces to the element end forces in globalcoordinates and accounts for the effect of element loading. The equilibrium matrix is made upof at least the product . The transformation of the dependent element forces in 6.4 to the globalcoordinate system includes at least the rotation matrix . After substituting 6.5 into the assemblyoperation in 6.1, the global resisting force vector is

(6.6)

For convenience of later discussion in this chapter, the basic forces of all elements are collected into asingle vector

Q

. After noting that the assembly operation in 6.6 only affects the rows of

p

and thus onlythe rows of and , 6.6 can be written in the compact form,

(6.7)

The structure equilibrium matrix

B

results from the assembly of the rows of while the columns arecollected from each element. Partitioning 6.7 into free and restrained DOFs, as indicated in 6.1, theequilibrium of the structure is represented by

(6.8)

Equation 6.8 is very significant in structural analysis because it must be satisfied for any frame element,made of linear or nonlinear material, under the limitation that equilibrium is satisfied in the undeformedconfiguration. The number of equilibrium equations in the free partition,

nf

, is the number of rows in, and the number of columns is the number of unknown basic forces,

nq

. For a statically determinatestructure,

nf

and

nq

are equal, and

B

f

is square and invertible, if the structure has a stable equilibriumconfiguration. Hence, for a statically determinate structure, given the applied forces at the free DOFs,

, we can solve the first partition in 6.8 for the unknown basic forces

Q

. If the support reactions aredesired, the second partition in 6.8 can be evaluated for .

For a statically indeterminate structure the number of unknown basic element forces,

nq

, is greaterthan the number of equilibrium equations,

nf

, at the free DOFs. Thus, the size of the matrix givesthe degree of static indeterminacy NOS =

nq – nf

. In statistically indeterminate structures, the equilibriumequations must be satisfied, but they are not sufficient to give a unique solution.

6.2.3 Lower Bound Theorem of Plastic Analysis

Since structures designed to resist earthquakes are rarely statically determinate, the most significantapplication of the structural equilibrium equations in 6.8 is for plastic analysis (Livesley, 1975). Assumingperfectly plastic material response requires that the basic element forces satisfy the plastic condition

, where are the plastic capacities of the elements. The applied forces at the free DOFs in 6.8are written as the product of a load factor and a reference force vector that gives the distributionof the applied loads, such as the equivalent lateral earthquake loads. The lower bound theorem of plasticanalysis states that the collapse load factor is the largest load factor that satisfies the equilibriumequations in 6.8 and the plastic condition. We can write this as follows:

p b bq b p= +r r w p b b bq b b p= +j r j r w

p b q p b b b b p b b p= + = =g w g j r w j r wwith and

bg

pw bg

b br pw

br

P p b q pA Ar g w= = +( )el

el

el

el el el( ) ( ) ( ) ( )

bg pw

P BQ Pr w= +

bg

P

P

B

BQ Pf

d

f

dw

ÊËÁ

ˆ¯

=ÊËÁ

ˆ¯

+

B f

Pf

Pd

B f

Q Q£ p Qp

l Pref

l c


Methods of Analysis for Earthquake-Resistant Structures 6-11

(6.9)

Considering that the plastic capacity may be different under positive basic forces than under negativeones, and collecting the unknowns of the problem and Q into a single vector, the equations in 6.9are written in the compact form of a linear programming problem:

(6.10)

is the plastic capacity for a positive basic force and is the capacity for a negative basic force,respectively (both in absolute value), and I is the nq ¥ nq identity matrix. The linear programmingproblem in 6.10 can be readily solved using the simplex method with widely available mathematicalsoftware packages such as Matlab„ or Mathcad„.

The solution of the linear programming problem in 6.10 yields a unique collapse load factor according to the lower bound theorem of plastic analysis. Even though the collapse load factor is unique,the basic forces at collapse are unique only if a complete collapse mechanism forms. This requiresthat NOS+1 basic forces reach the plastic capacity (in other words, that NOS+1 plastic hinges form),where NOS is the degree of static indeterminacy. This requirement derives from the equilibrium equationsin 6.10, i.e., from the requirement that

(6.11)

There are nq+1 unknowns and nf available equations of equilibrium in 6.11. Because NOS = nq – nf,there are NOS+1 more unknowns than available equations of equilibrium. With the value at NOS+1basic forces equal to the corresponding plastic capacity at collapse, there are as many unknowns in 6.11as the number of equilibrium equations. A partial mechanism forms if fewer than NOS+1 basic forcesreach the plastic capacity. In this case, there exist an infinite number of combinations of the remainingbasic forces that satisfy the equilibrium equations in 6.11. A unique solution can only be obtained withan additional assumption about the force–deformation behavior of the elements before reaching theplastic capacity, which we will pursue later in this chapter.

6.3 Geometric Compatibility

The statement of geometric compatibility is analogous to the process of establishing the equilibriumequations of the structural model. Each node can undergo three translations and three rotations in the3d case; for the 2d case each node undergoes two translations and one rotation. The displacements ofall nodes in the model are collected in the displacement vector U. From the compatibility betweenelements and nodes, the displacements at the end of an element are equal to the corresponding DOFdisplacements at the nodes (see Figure 6.6). This correspondence between global and element DOFs isprovided again by the id-array of each element, and the compatibility can be written symbolically as

(6.12)

Equation (6.12) indicates that an extraction from the global displacement vector U of only the degreesof freedom corresponding to the entries in the id-array of the element el takes place. The length of thevectors in 6.12 is, consequently, equal to the number of element DOFs.

l l lc = = £max for andref f pP B Q Q Q

l

ll l l

c = [ ]ÊËÁ

ˆ¯

-[ ]ÊËÁ

ˆ¯

=-

È

ÎÍ

˘

˚˙

ÊËÁ

ˆ¯

£Ê

ËÁ

ˆ

¯˜

+

-max 1 0Q

P BQ

00 I

0 I Q

Q

Qfor andref f

p

p

Qp+ Qp

–

l c

Qc

P BQ

0ref fc

-[ ]ÊËÁ

ˆ¯

=l c

u U( )elid=


6-12 Earthquake Engineering: From Engineering Seismology to Performance-Based Engineering

As described in Section 6.1.3, the displacement vector for the structure is partitioned into free DOFs, which are unknown at the start of the analysis, and the displacements at the restrained DOFs ,

which are specified.

6.3.1 Displacement–Deformation Relationship under Large Displacements

Figure 6.7 shows a two-node frame element in the undeformed and deformed configuration under givenend displacements. Since the rigid-body displacement of the element does not generate element defor-mations, Figure 6.7 shows three stages for representing the relationship between displacement anddeformation. Figure 6.7a provides the overview of the entire process and identifies the rigid-body trans-lation of the element. The relative displacements and in the local reference system areconvenient for describing the relative position of node j with respect to node i. This relative translationresults in an extension of the element in Figure 6.7b. As long as the ends follow the rigid-body rotationof the element axes through the angle , no other deformation arises in Figure 6.7b, as illustrated bythe rotation of the black squares representing the end nodes. Since the end rotations are independentDOFs, each end is subjected to an additional rotation past angle and the element can deform, asshown in Figure 6.7c. The rotation of the end tangent to the deformed shape relative to the chord linefor the element results from flexural deformation. There are two rotations caused by the flexural defor-mations, and , in Figure 6.7c, one at each element end. Counterclockwise rotations measuredfrom the chord to the tangent are positive, consistent with the sign convention for the end moments.

With the definition of element deformations, v, it is now possible to derive the relationship betweenthe element deformations and the end displacements, , in the local coordinate system. Using Figures6.7b and 6.7c, the element deformations for the general case of large displacements and moderatedeformations can be given as

FIGURE 6.6 Compatibility between nodal and element end displacements.

FIGURE 6.7 Deformed configuration of two-node one-dimensional element under large displacements.

i jx

y

bu-3

u-6

v2

v3

L

(c)

i

i

j

j

x

yDu-y = u-5 - u-2

Du-y = u-5 - u-2

Du-x = u-4 -u-1

Du-x = u-4 - u-1

b

position under rigid body translation

Ln

v1

(b)

u4

u5

u1

u2

X

Y

x

y L

L

Ln

u3

u6

b

(a)

U f Ud

Dux Duy

v1

b

b

v 2 v3

u



(6.13)

(6.14)

In 6.13 we have used the definition of the Lagrange strain for the length change of the element. In atypical structural analysis for earthquake engineering, the engineering strain is a sufficient approximation,so the extension is . The arctan function when expanded by a Taylor series about the point

gives for the chord rotation:

Similarly, expansion of the first equation in 6.13 gives the change in length of the element:

Factoring L out on the right-hand side of the above expression gives

(6.15)

In earthquake engineering applications the axial strain has values of order10-3 to 10-2. In this

case the first quadratic term on the right-hand side of 6.15 is of order 10-6 to 10-4 and can be neglected.

The relative transverse displacement (commonly called the drift), , is of order10-2 to 10-1, so that

the second quadratic term in 6.15 can be of the same order as the linear term. In such situations theelement deformation should be approximated with

(6.16)

The relation between the displacement components in the global and the local coordinate systems usesthe rotation transformation for the translation components, whereas the rotations in the plane are notaffected. The rotation angle for the transformation of the translation components corresponds to theangle of the undeformed element x-axis relative to the global X-axis (note that the positive element x-axis points from node i to node j). The translation components at each element end are transformedindependently. The transformation of the end displacements is expressed symbolically by .

v

v u

v u

12 2

2 3

3 6

1

2= -( )= -

= -

LL Ln

b

b

with L L

L

n x y

y

x

= +( ) + ( )=

+

D D

DD

u u

u

u

2 2

b arctan

v1 = -L Ln

D Du ux y= =0 0,

b = - +ÈÎÍ

˘˚

D Du uy x

L L1 ...

v uu u

1

2 21

2

1

2= + +D

D Dx

x y

L L

vu u u

1

2 2

1

2

1

2= + Ê

ËÁˆ¯

+Ê

ËÁˆ

¯

È

Î

ÍÍ

˘

˚

˙˙

LL L L

x x yD D D

DuxL

Duy

L

vu u

1

2

1

2= +

Ê

ËÁˆ

¯

È

Î

ÍÍ

˘

˚

˙˙

LL L

x yD D

u a u= r



6.3.2 Linear Approximation of Displacement–Deformation Relation

In the cases where the relative transverse displacement of the element , , is of order or less,

the quadratic term in 6.16 can be neglected and a linear compatibility relation remains . In the

expression for the angle the term is very small relative to unity and can be neglected in any

case. With these approximations, the compatibility relations between element deformations and enddisplacements in the local coordinate system become linear:

(6.17)

Comparing 6.17 with 6.3, we note that the compatibility matrix a is the transpose of the equilibriummatrix b. The same relation holds for matrices and . If the element has rigid-end offsets, thecompatibility matrix between the displacements at the ends of the deformable portion of the elementand those at the ends of the complete element is similarly the transpose of the equilibrium matrix .The generality of this observation will be proven in the following section.

After combining the compatibility relationships, the element deformations can be expressed in termsof the element end displacements in the global coordinate system:

(6.18)

The compatibility transformation matrix is the transpose of the equilibrium matrix in 6.5. With6.12 the basic element deformations can now be expressed in terms of the global displacement vectorcomponents that correspond to the element as .

It is worthwhile to collect the element deformations into a single vector V for the structural model.In this process the element compatibility matrices can be combined with the aid of the id-array to givethe structural compatibility matrix A. After this combination, the compatibility between the elementdeformations and the free and restrained DOFs at the nodes assumes the following form:

(6.19)

From the process of forming the structural compatibility matrix A, we observe that it is the transposeof the equilibrium matrix B in 6.7 and 6.8. The same is obviously true for submatrices and and

and Bd, respectively. In practice, only one of these matrices need be established and the other isobtained as its transpose. In the following we assume that the compatibility matrix A is assembled andthe equilibrium matrix is obtained by transposition. Thus, we alternate our reference to the equilibriummatrix as B or , as circumstances require.

Duy

L 10 2-

v u1 = D x

b DuxL

v u u u

v uu

uu u

v uu

uu u

v

v

v

v

1 4 1

2 3 35 2

3 6 65 2

1

2

3

1 0 0 1 0 0

01

1 01

0

01

0 01

1

= = -

= - = --

= - = --

=

Ê

Ë

ÁÁÁÁÁ

ˆ

¯

˜˜˜˜

=

-

-

-

È

Î

ÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙

D

D

D

x

y

y

L L

L L

L L

L L

or

uu

u

u

u

u

u

au

1

2

3

4

5

6

Ê

Ë

ÁÁÁÁÁÁÁÁÁÁÁÁ

ˆ

¯

˜˜˜˜˜˜˜˜˜˜˜˜

=

a r br

a j

b j

v a u aa a u a u= = =r j g

ag bg

v a u a U= =g g id

V A U A AU

U= = [ ]Ê

ËÁˆ¯f d

f

d

A f B f

Ad

AT



6.3.3 Compatibility Relationship for Elements with Moment Releases

It is worth looking in more detail into the case of a frame element with a moment release at one endbecause of later consideration of plastic hinges. A moment release at one end is achieved by introducinga hinge at the end. Referring to Figure 6.8 the element deformations are measured from the chord tothe tangent at the element side of a hinge at one end or the other. If an end moment is released to zero,the rotations at the ends are no longer independent, and we will show later that for a prismatic, linearelastic beam the deformation at a moment release is one half the value of the deformation at the oppositeend, but of opposite sign. The total deformations at the element ends are measured from the chordto the tangent at the node side of the hinge. These are the sum of the element deformations v and thehinge deformations . The following relationships for a moment release at end i or end j can beestablished from Figure 6.8:

For convenience, the geometric relationships for the two cases can be combined into a single relation byintroducing a binary variable as moment release code for each end. Variable mr assumes the value 0when the end is continuous, and the value 1 when a release is present at the corresponding end. Withthis variable we can write the above transformation relations in a compact form,

which now holds also for the case without moment releases, a moment release at either end and momentreleases at both ends. After combining the flexural deformations with axial deformation, which is unaf-fected by the moment releases, the basic compatibility relationship is

(6.20)

FIGURE 6.8 Flexural deformations for element with moment release.

i j

L

v3

v2 v3

v2 v3

v

v

v h

v

v

v

v

v

v

v

v2

3

2

3

2

3

2

3

1 01

20

01

20 1

ÊËÁ

ˆ¯

= -È

ÎÍÍ

˘

˚˙˙

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

= -È

ÎÍÍ

˘

˚˙˙

ÊËÁ

ˆ¯

˜

˜

˜

˜for hinge at end j for hinge at end i

v

v

v

v2

3

2

3

11

21

1

21 1

ÊËÁ

ˆ¯

=- - -( )

- -( ) -

È

Î

ÍÍÍ

˘

˚

˙˙˙

ÊËÁ

ˆ¯

mr mr mr

mr mr mr

i j i

i j j

˜

˜

v a v a= = - - -( )- -( ) -

È

Î

ÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙

h hwith˜

1 0 0

0 11

21

01

21 1

mr mr mr

mr mr mr

i j i

i j j



A precise definition of the hinge rotation will be useful when plastic hinges are considered. The hingerotations are the difference between the deformations and the element deformations v. Conse-quently, hinge rotations are measured from the tangent at the element side of the hinge to the tangentat the node side. We can, therefore, write the hinge rotation as

where I is the identity matrix, and the total end deformations are given by 6.18. In conclusion, thegeometric compatibility relationship covers all cases of frame elements with or without endmoment releases. This result will be used in the subsequent discussion of plastic hinge rotations and inthe development of the force–deformation relationship of elasto-plastic elements.

6.4 Equilibrium in Deformed Configuration

With the definition of the element compatibility relationship under large displacements we can nowestablish the equilibrium relationship for an element in the deformed configuration. Assuming that thebasic forces act on the deformed element, acts always along the deformed element chord and itchanges orientation as the element deforms. This approach is known as corotational formulation(Crisfield, 1990, 1991), but other names such as member-bound reference system or physical coordinateshave also been used (Argyris, et al., 1979, Elias, 1986).

The element end forces in the reference frame of the deformed element can be expressed in terms ofthe basic forces by satisfying the equilibrium equations of the element-free body in the deformedconfiguration, as illustrated in Figure 6.9. These equations are identical to those in Figure 6.4c except forthe fact that the deformed element length is used in place of L. Subsequently, all forces are transformedfrom the member-bound coordinate system to the local coordinate system by a rotation through angle

, defined as the angle between the chord of the deformed element and the undeformed position (Figure6.9). The equilibrium equations are

(6.21)

FIGURE 6.9 Equilibrium of frame element in deformed configuration.

i jq1

q2

q3q1

Ln

q2 + q3 Ln

q2 + q3

x

b

L

Ln

p-3p-1

p-5

p-6 p-4

p-2

Du-y = u-5 - u-2

Du-x = u-4 - u-1

v h v

v v v I a vh h= - = -( )˜ ˜

vv a a u= h g

q1

Ln

b

p

p

p

p

p

p

p

u u u

u u u

u u u

u u

=

Ê

Ë

ÁÁÁÁÁÁÁÁÁÁ

ˆ

¯

˜˜˜˜˜˜˜˜˜˜

=

-+( )

- -

-+ +

+

-+

1

2

3

4

5

6

2 2

2 2

2 2

0 1 0

L

L L L

L

L

L

L

L

L

L L L

L

L

x

n

y

n

y

n

y

n

x

n

x

n

x

n

y

n

y

n

y

n

D D D

D D D

D D D

D D xx

n

x

nL

L

L2 2

1

2

3

0 0 1

-+

È

Î

ÍÍÍÍÍÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙˙˙˙˙˙

Ê

Ë

ÁÁÁÁ

ˆ

¯

˜˜˜

=

Du

q

q

q

b qu



This relationship is equivalent to 6.3 except that equilibrium is now satisfied in the deformed configu-ration. The subscript of the equilibrium matrix is a reminder that the equilibrium depends on theend displacements. For applications in earthquake engineering analysis, it is reasonable to approximatethe equilibrium matrix in 6.21 by expanding the terms of in a Taylor series and including only termsthat have a consistent order of magnitude. The expansion of the terms in the equilibrium matrix is

Recalling the order of magnitude of the terms and in comparison with unity, we concludethat we can neglect the factors in square brackets and the quadratic term in the first expression. Withthese approximations the equilibrium relations in 6.21 simplify to

(6.22)

The relationship in 6.22 is a consistent representation of the effect of large displacements on the equilibriumof a frame element. For many models undergoing moderate deformations, it is reasonable to further simplify

6.22 by neglecting the contribution of the shear force to the axial force, because of the small

magnitude of shear relative to the axial force, and because the transverse deformation, , does not

exceed 0.1 in most cases (about 10% drift of the element). With this approximation 6.22 simplifies further to

(6.23)

bu

bu

L

L L

L L L L

L

L L L L

L L

x

n

y

y

n

y x y

x

n

x y

y

n

y

+( )ª -

Ê

ËÁˆ

¯

ª - -Ê

ËÁˆ

¯

È

Î

ÍÍ

˘

˚

˙˙

+( )ª - -

Ê

ËÁˆ

¯

È

Î

ÍÍ

˘

˚

˙˙

ª -

D D

D D D D

D D D

D D D

u u

u u u u

u u u

u u u

1

11

2

11

1 2

2

2

2

2

2 2xx y

L L-

Ê

ËÁˆ

¯

È

Î

ÍÍ

˘

˚

˙˙

Du2

DuxL

Duy

L

p

p

p

p

p

p

p

u u

u

u u

u

=

Ê

Ë


ˆ

¯

˜˜˜˜˜˜˜˜˜˜

=

- - -

-

- -

È

Î

ÍÍÍÍÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙˙˙˙˙

1

2

3

4

5

6

2 2

2 2

1

1 1

0 1 0

1

1 1

0 0 1

D D

D

D D

D

y y

y

y y

y

L L

L L L

L L

L L L

qq

q

q

b q

1

2

3

Ê

Ë

ÁÁÁÁ

ˆ

¯

˜˜˜

= u

q q2 3+L

Duy

L

p

p

p

p

p

p

p

u

u

q

q

q

b=

Ê

Ë


ˆ

¯

˜˜˜˜˜˜˜˜˜˜

=

-

-

- -

È

Î

ÍÍÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙˙˙

Ê

Ë

ÁÁÁÁ

ˆ

¯

˜˜˜

=

1

2

3

4

5

6

1

2

3

1 0 0

1 1

0 1 0

1 0 0

1 1

0 0 1

D

D

D

y

y

L L L

L L L

P qq



The approximate equilibrium equation in 6.23 is often used in the so-called P-D analysis of structures.The transformation of the end forces from the local to the global coordinate system and any further

equilibrium transformations addressed earlier are not affected by the equilibrium transformation of thebasic system in Figure 6.9. Thus, the general case of element equilibrium can be expressed as

By selecting the transformation matrix depending on the needs of the analysis, one can accommodatelarge displacements with from 6.21, geometry with from 6.23 and linear geometryunder small displacements with from 6.5.

6.5 Principle of Virtual Work

The virtual work principles are convenient representations of the equations of equilibrium and theconditions of geometric compatibility. With virtual work we can express the equilibrium and compati-bility equations, which are vectorial, by a scalar work equation. These principles are derived by startingfrom either the equilibrium or compatibility equations of an element, converting these into integral formby multiplication with a virtual field, performing integration by parts that introduces the boundary termsand then summing up all element contributions. In the following we only state the final result of thederivation.

6.5.1 Virtual Work Principles for the Structure

The principle of virtual displacements is equivalent to the satisfaction of the equations of equilibriumin the structure. It states that the work done by a set of virtual displacements on the external forces(external work) is equal to the work done by the compatible virtual deformations on the element forces(internal work), if the external forces are in equilibrium with the element forces, and vice versa. In theabsence of element loads the principle of virtual displacements is

(6.24)

where the virtual displacements and deformations satisfy the compatibility requirement as in 6.19. Substituting this condition for the virtual displacements in 6.24 gives

(6.25)

If 6.25 holds for arbitrary virtual displacements , then and the converse is also true.Therefore the principle gives the equilibrium equations, in 6.7, demonstrating that theequilibrium matrix is equal to the transpose of the compatibility matrix, .

The principle of virtual forces is equivalent to the satisfaction of the conditions of geometric compat-ibility. It states that, if the deformations are compatible with the displacements, then the complementarywork of a set of virtual external forces on the displacements is equal to the work of the virtual internalforces that are in equilibrium with the external forces on the deformations. The reverse is also true. Incompact form the principle of virtual forces is

(6.26)

Substituting the equilibrium requirement for the virtual forces , into 6.26 gives

p

p b b b q= j r u

bu

bu P - D b bu P= Db bu =

d dU P V QT T=

dU dVd dV A U=

d d d dU P V Q U A Q U P A QT T T T T T= = Æ -( ) = 0

dU P A Q 0- =T

P A Q 0- =T

B A= T

d dP U Q VT T=

d dP B Q=



which provides the geometric compatibility requirement in 6.19 along with the condition that .

6.5.2 Virtual Work Principles for an Element

We now turn our attention to an individual frame element, see for example, Figures 6.4 and 6.7. Theexternal work is done by the end forces on the corresponding displacements, whereas the internal workis done by the basic forces on the corresponding deformations. Without further derivation, we state thisrequirement as

(6.27)

The condition of geometric compatibility of the virtual displacements is , which upon sub-stitution into 6.27 gives

(6.28)

Comparing 6.28 with the element equilibrium equations in 6.5 shows that .We can generalize this fact by stating that if a set of displacements transforms from one system to

another according to the relationship , then the forces corresponding to these displacementstransform according to the contragradient relationship . Similarly, if a set of forces transformsfrom one system to another according to , then the corresponding displacements transformaccording to . Consequently, we only need to establish either the force or displacement trans-formation relationship from one coordinate system to another using the equilibrium or compatibilityconditions, respectively.

Returning now to the virtual work statement for a single element, we can derive the internal workfrom the integral of the stress product with the corresponding virtual strains over the element volume V:

(6.29)

where eeee is a vector containing the components of the strain tensor and ssss is a vector with the correspondingcomponents of the stress tensor arranged in the same order. In many applications of nonlinear structuralanalysis, we limit ourselves to the internal work of the axial stress and shear stress on the axialstrain and shear strain , respectively. In this case 6.29 reduces to

(6.30)

From the principle of complementary virtual work for the element we obtain the corresponding com-patibility relationship for the element:

These virtual work statements will be used in the next section for deriving the force–deformationrelationships for linear and nonlinear elements.

d d d d dP U Q V Q B U Q V Q B U VT T T T T T T= Æ = Æ -( ) = 0

B AT =

d du p v qT T=

d dv a u= g

d d d du p v q u a q u p a q p a q 0T T TgT T

gT

gT= = Æ -( ) = Æ - =0

b ag gT=

v a u= g

p a q= gT

p b q= g

v b u= gT

d dv qT = Ú ee ssT

V

dV

s x te x g

d de s dg tv qT = +( )Ú x x

V

dV

d ds e dt gq vT = +( )Ú x x

V

dV



6.5.3 Upper Bound Theorem of Plastic Analysis

We return to the problem of plastic analysis and complement the earlier development in Section 6.2.3with the determination of the plastic collapse load factor by the upper bound theorem of plastic analysis.To state the latter it is necessary to formulate the external and internal plastic work past the point ofmaximum load. The assumption of perfect plasticity leads to the conclusion that only plastic deformationincrements arise in the elements once a collapse mechanism has formed. The plastic deformation incre-ments must satisfy the conditions of geometric compatibility for the free DOFs, which is written in aform similar to 6.19:

(6.31)

The external plastic work increment is . The internal work consists of the product of the plasticcapacities of the elements and the plastic deformation increments. Because the plastic capacities have beendefined as absolute values in 6.9, the plastic deformation increments are defined as if ,otherwise it is zero; if , otherwise it is zero. With this definition the internal plasticwork increment becomes and 6.31 changes to , noting more-over that and .

The upper bound theorem of plastic analysis states that the collapse load factor is the smallestload factor satisfying the condition of incremental plastic work and geometric compatibility. This is statedin a compact form as

(6.32)

Equation 6.32 can be written in the standard form of linear programming by constraining and noting that the unknowns of the problem are now DUf, DVp

+ and DVp– (Livesley, 1975). With

these considerations, 6.32 is written in a compact form by collecting the unknowns in a single vector:

(6.33)

Comparing 6.33 with 6.10 and recalling that , we conclude that the upper bound theorem ofplastic analysis in 6.33 is the dual problem of linear programming to the lower bound theorem in 6.10,so that the solution of one satisfies the other. Consequently, the collapse load factor from either 6.10 or6.33 is the unique solution of the plastic analysis problem. It is also interesting to observe from 6.31 thatthe columns of the compatibility matrix represent the independent collapse mechanisms of the structuralmodel. The dependent collapse mechanisms can be obtained by linear combination of these columns.

6.5.4 Example of Plastic Analysis

The two-story frame in Figure 6.10a (Horne and Morris, 1982) illustrates the concepts of plastic analysis.With the assumption that axial deformations are negligibly small, there are ten free DOFs, as shown inFigure 6.10b. The plastic moment capacities of the columns and girders are enclosed in a circle in Figure6.10a and the reference loading is also shown.

D DV A Up f f=

l P UrefT

fD

D DV Vp p+ = DVp ≥ 0

D DV Vp p- = - DVp £ 0Q V Q Vp p p p

+ + - -+D D D D DV V A Up p f f+ -- =

DVp+ ≥ 0 DVp

- ≥ 0l c

l l lc = = +

- = ≥ ≥

+ + - -

+ - + -

min

,

for

and

refT

f p p p p

p p f f p p

P U Q V Q V

V V A U V V

D D D

D D D D D0 0

P UrefT

fD = 1

l c = [ ]Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜ - -

È

ÎÍÍ

˘

˚˙˙

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜

=ÊËÁ

ˆ¯

≥ ≥

+ - +

-

+

-

+ -

min

,

0 Q Q

U

V

V

P 0 0

A I I

U

V

V0

V V

p p

f

p

p

refT

f

f

p

p

p p

for

with

DDD

DDD

D D

1

0 0

A Bf fT=



The first step consists in setting up the compatibility matrix relating the element deformations tothe DOFs of the model. The model has eight elements with two deformations each, since axial deforma-tions are neglected. Thus, the compatibility matrix has 16 rows and 10 columns. The deformed shapesfor the vertical and horizontal translation DOFs and for one rotation DOF are shown in Figure 6.11.

(a) (b)

FIGURE 6.10 (a) Two-story frame. (b) Free global DOFs without axial deformations.

FIGURE 6.11 Typical deformed shapes for global DOFs of two-story frame.

30 kN

36 kN40 kN

20 kN

4 m 4 m

4 m

4 m170 170

100

200

80 80

123

45

678

910

7

1

2

1 1 1 4

1 19

5

A f



Denoting the story height by h and the girder span by 2l, the compatibility matrix is

The columns of the compatibility matrix represent the independent collapse mechanisms of the frame.Columns 1, 3, 5, 6, 8, 10 represent the joint mechanisms, columns 2 and 7 the beam mechanisms andcolumns 4 and 9 the story mechanisms. Figure 6.12 shows the beam mechanisms and the lower storymechanism in which a gray circle indicates a plastic hinge. Plastic rotations are measured from the tangentat the element side of the hinge to the tangent at the node side. Consequently, the lower girder rotationat the left end in Figure 6.12a is negative. The similarity of the shape for the mechanism with thecorresponding deformed shape in Figure 6.11 is evident.

The equilibrium equations of the problem are . We can elect to solve either thelower bound problem in 6.10 or the upper bound problem in 6.33. With either approach the collapseload of the frame is . Only six plastic hinges form at collapse, as shown in Figure 6.13b,indicating a partial collapse mechanism since NOS = 6. Consequently, there are more unknown Q valuesthan the available equations of equilibrium in 6.11, and the moment distribution in Figure 6.13a is notunique. In this regard, the double hinge at midspan of the upper girder counts as one.

FIGURE 6.12 A few independent collapse mechanisms.

7

2

4

(a) (b)

A f =

--

--

0 0 0 1 0 0 0 0 0 0

1 0 0 1 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0

0 1 1 0 0 0 0 0 0 0

0 1 1 0 0 0 0 0 0 0

0 1 0 0 1 0 0 0 0 0

0 0 0 1 1 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0

1 0 0 1 0 0 0 0 1 0

0 0 0 1 0 1 0 0 1

h

h

l

l

l

l

h

h

h h

h hh

l

l

l

l

h h

h h

0

0 0 0 0 0 1 1 0 0 0

0 0 0 0 0 0 1 1 0 0

0 0 0 0 0 0 1 1 0 0

0 0 0 0 0 0 1 0 0 1

0 0 0 1 0 0 0 0 1 1

0 0 0 1 1 0 0 0 1 0

--

--

È

Î

ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙

lP B Q A Qref f fT= =

l c = 2 50.



6.6 Force–Deformation Relationships

The previous sections have developed the equilibrium and compatibility relationships for a frame ele-ment, which only depend on the element geometry and not on the constituent materials of the element.It is now necessary to relate the element basic forces to the corresponding deformations. To do so in asystematic manner we start with the consideration of the section response and show how it is establishedby integration of material response. Subsequently, we briefly describe a few material models used in theexamples of this chapter.

6.6.1 Section Response

We return to the virtual work statement for an element in 6.30 and rewrite the integral over the elementvolume as integration over the section at a location x followed by integration over the element length:

FIGURE 6.13 Moment distribution and mechanism at incipient collapse.

(a) Moment distribution

(b) Collapse Mechanism



(6.34)

The strain and stress are functions of the position x along the element axis and the position within the crosssection specified in local coordinates y and z. The axial strain at point M in Figure 6.14 can be written asthe product of two functions,

(6.35)

where are the section deformations and represents the strain distribution at section x.When shear deformations are small, Bernoulli’s assumption of plane sections remaining plane proves anexcellent approximation. In this case the strain distribution at section x is

(6.36)

and the section deformation vector, , consists of the axial strain at the coordinate origin , thecurvature about the z-axis and the curvature about the y-axis . Figure 6.14 shows an arbitrarysection. The sign convention in 6.36 follows the right-hand rule for rotation and the definition of tensilestrain as positive. The integration of the virtual work expression over the cross-section area A becomes

and the virtual work for the element in 6.34 can be written as

(6.37)

The work terms corresponding to the section deformations are the section forces definedaccording to

(6.38)

FIGURE 6.14 Cross section with coordinate axes.

y

z

M

d de s dg tv qT = +( )È

ÎÍÍ

˘

˚˙˙ÚÚ x x

AL

dA dx

e x x y z y z x( , , ) ( , ) ( )= a es

e( )x a s( , )y z

a s( , )y z y z= -[ ]1

e( )x e a

k z k y

de s d s d s ddA x y z dA x y

z

dA x x

A A A

Ú Ú Ú= = -Ê

Ë

ÁÁ

ˆ

¯

˜˜ =e a e e sT

sT T T( ) ( , ) ( ) ( ) ( )

1

d dv q e sT T= Ú ( ) ( )x x dx

L

e( )x s( )x

s( )

( )

( )

( )

x y

z

dA

N x

M x

M xA

z

y

= -Ê

Ë

ÁÁ

ˆ

¯

˜˜ =

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜Ú

1

s



The right-hand side of 6.38 is the standard definition for the axial force and bending moments aboutthe z- and y-axes, respectively.

We define the section stiffness matrix ks(x) as the partial derivative of the section forces s(x) withrespect to section deformations . With this definition and the rules of differentiation, the sectionstiffness is

(6.39)

Equation 6.39 uses 6.35 for the derivative of the axial strain with respect to . The partial derivativeof stress with respect to strain is the tangent modulus, Et, of the material stress–strain relationship.

For the special case of linear elastic material with modulus E, 6.39 gives the standard definition forthe section stiffness under axial and flexural behavior,

(6.40)

in which Q denotes the first moment and I the second moment of area, respectively. For linear elasticmaterial the section force–deformation relationship becomes

(6.41)

It is possible to select the origin at the centroidal axis of the section and the orientation of the y–z axesalong the principal axes, which renders the off-diagonal terms of the section stiffness in 6.40 equal tozero. Although the centroidal axis is a useful reference point for a homogeneous section with linear elasticmaterial, it is meaningless for the general case of a section with nonlinear materials.

In the general case of a nonlinear stress–strain relationship it is not possible to evaluate the integralsin 6.38 and 6.39 in closed form. Therefore, numerical integration is needed, which gives the value of anintegral as a summation,

(6.42)

in which is the function to be integrated, nIP is the number of integration points and is theweight at integration point i. Different integration rules can be used in 6.42, but discontinuities associatedwith the stress distribution in 6.38 and the tangent modulus in 6.39, particularly under cyclic loadreversals, favor low order integration schemes such as midpoint, trapezoidal or Simpson’s rule. Theaccuracy of integration is improved with a larger number of integration points. The midpoint rule is themost common integration scheme in the application of nonlinear analysis in earthquake engineeringand gives rise to the name layer model for y-integration or fiber model for y–z integration. It is worthnoting that the midpoint rule exactly integrates linear polynomials. Consequently, the quadratic stiffnessterms in 6.39 are not accurately represented even for the linear elastic case. For the typical number ofintegration points the error is, however, very small.

We conclude this section by summarizing that the cross-section response can be obtained by integrationof the stress–strain response of the materials. A kinematic assumption about the strain distribution is

e( )x

ks

e es

A A A

xx

xy

zx

dA y

z

y z dA

y yz

y y z

yz z z

dA( )( )

( ) ( ) = ∂

∂= -

Ê

Ë

ÁÁ

ˆ

¯

˜˜∂∂

∂∂

= -Ê

Ë

ÁÁ

ˆ

¯

˜˜∂∂

-[ ] = ∂∂

- --

-

È

Î

ÍÍÍ

˘

˚

˙˙˙

Ú Ú Ú1 1

1

12

2

se

e se

se

e( )x

ks ( )x E

y z

y y yz

z yz z

dA E

A Q Q

Q I I

Q I IA

z y

z z yz

y yz y

=-

- --

È

Î

ÍÍÍ

˘

˚

˙˙˙

=

È

Î

ÍÍÍ

˘

˚

˙˙˙Ú

12

2

s k e( ) ( ) ( )x x x= s

g y z dA w g y z

A

i i i

i

nIP

( , ) ( , )Ú Âª=1

g y z( , ) wi



typically the starting point. In the general case of including all components of the strain tensor eeee(x, y, z)we write

(6.43)

Definitions for section deformations and corresponding forces need to be generalized accordingly. ∂s/∂eeeeis, in general, a 6 ¥ 6 matrix representing the tangent material stiffness. The examples of this chapter arelimited to uniaxial material response.

Before completing the discussion of force–deformation relationships, we mention that the sectionresponse can be directly defined in terms of explicit or implicit section force–deformation relations. Suchan approach is based on the extension of plasticity theory to section force resultants and deformations(McGuire et al., 2000). While this approach is an excellent choice for homogeneous sections of metallicmaterial, it is doubtful whether it constitutes a robust alternative to the integration of material responsefor sections composed of several materials. This is particularly the case under complex interactions ofthe constituent materials that may lead to softening response, as is the case for reinforced concretesections.

6.6.2 Material Models

A key ingredient for establishing the section response with 6.43 is the constitutive model of the material.Under the assumptions in the preceding section, uniaxial material models suffice for many applicationsin earthquake engineering analysis. Thus, relatively complex uniaxial material relations can be deployedwithout difficulty. However, the high computational cost of integrating a complex material response,usually limits the selection to a few relatively simple hysteretic models.

For applications in performance-based earthquake engineering it is important to distinguish betweenpath-dependent and path-independent material models. In a path-independent model the current stressis a function of current strain only. Thus, the material follows the same path whether it is loading orunloading. In static pushover analysis of structures this may be sufficient, if one can assume that limitedunloading, if any, will take place. This is often the case. In cyclic static or dynamic analysis a path-depen-dent material model is required. In this the current stress depends on the current strain and several othervariables, such as the strain history and internal variables, that describe the state of material damage.The more complex the path of loading–unloading of a material model, the higher the number of internalvariables required in its description with a consequent increase in computational cost.

For the purposes of this chapter we limit ourselves to a brief description of three relatively simplematerial models which are used in the examples of this chapter: a bilinear elasto-plastic model withkinematic and isotropic hardening and a more involved version that includes the Bauschinger effect.Both models are suitable for describing the hysteretic behavior of steel. Finally, we briefly describe asimple hysteretic model for concrete.

Figure 6.15 shows a bilinear elasto-plastic model with kinematic and isotropic hardening. The isotropichardening depends on the amount of plastic strain in the opposite stress direction, which is why it isonly evident under compressive stress. The bilinear model is a good representation of the behavior ofmetals and is often used in earthquake analysis when the Bauschinger effect is not important for thesimulations. It is computationally advantageous for its simplicity. When the Bauschinger effect is impor-tant, the stress–strain relation of Menegotto–Pinto in Figure 6.16 gives a very good representation of thematerial response (Menegotto and Pinto, 1973). It is particularly important for the computationaleconomy of analysis of frame structures that the model expresses stress directly as function of strain. The

section kinematics or compatibility

section equilibrium

section stiffness

s

sT

s sT

s

e( , , ) ( , ) ( )

( ) ( , )

( ) ( , ) ( , )

x y z y z x

x y z dA

x y z y z dA

A

A

=

=

= ∂∂

ÚÚ

a e

s a

k a a

s

se



FIGURE 6.15 Hysteretic bilinear stress–strain relationship.

FIGURE 6.16 Hysteretic steel stress–strain relation by Menegotto-Pinto.

FIGURE 6.17 Hysteretic concrete stress–strain relation.

−100

−80

−60

−40

−20

0

20

40

60

80

100

−0.005 0.000 0.005 0.010 0.015 0.020 0.025 0.030

Strain (in/in)

Str

ess

(ksi

)

−100

−80

−60

−40

−20

0

20

40

60

80

100

−0.005 0.000 0.005 0.010 0.015 0.020 0.025 0.030

Strain (in/in)

Str

ess

(ksi

)

0

1

2

3

4

5

6

7

−0.002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016

Concrete Strain (in/in)

Con

cret

e S

tres

s (k

si)



limitation of the model lies in its inability to reach the point of last unloading upon reloading in thesame stress direction. This is evident in Figure 6.16. Details of the model implementation and parameterselection can be found elsewhere (Filippou et al., 1983).

Figure 6.17 shows a simple hysteretic model for concrete. The monotonic behavior in compression isrepresented by a parabolic ascending curve followed by a linear descending curve to a residual stress of between10 and 20% of the compression strength. The slope of the descending curve can be adjusted to represent theeffect of concrete confinement provided by transverse reinforcement (Scott et al., 1982). The unloading–reload-ing path is also linear with decreasing modulus that follows the observations of an extensive experimentalstudy (Karsan and Jirsa, 1969). Several unloading–reloading cycles are shown in Figure 6.17. The model inFigure 6.17 does not account for the tensile strength of concrete and the effect of tension softening. On theother hand it is computationally very simple. More sophisticated models of concrete response under tensilestress are available at the expense of computational complexity (CEB, 1996). Their use is important when theprecracked and preyield response of reinforced concrete structures is of particular interest.

6.7 Frame Elements

After having established equilibrium, compatibility and section force–deformation relationships, we nowdevelop nonlinear frame elements with a range of applicability in structural analysis procedures forearthquake engineering. We discuss the advantages and limitations of each approach in the formulationof the frame elements. Emphasis is placed on a rigorous derivation of the element response and on thepresentation of the element state determination process, which consists in computing the basic forcesand the stiffness matrix that correspond to given element deformations.

6.7.1 Basic Relationships

The differential equations of equilibrium for a frame element in the undeformed configuration can bewritten, with reference to Figure 6.18, for axial and moment equilibrium, as

(6.44)

in which and are the axial and transverse components of the distributed element load, respec-tively. An important characteristic of frame elements is that, under linear geometry, the differential

FIGURE 6.18 Equilibrium in undeformed configuration of frame element.

frame element

x

applied external load wy(x)

L

x

i j

y

wy (x)

M (x + Dx)

V (x + Dx)V (x)

M (x)

Dx

x

wx

N (x)

Dx

N (x + Dx)

applied external load wx

∂∂

∂∂

N

xw x

M

xw x

x

y

+ =

- =

( )

( )

0

02

2

w x w y



equations in 6.44 can be solved independently of the displacements and of the material response. In theabsence of element loading the homogeneous solution of the differential equations in 6.44 gives a constantaxial force and a linear bending moment distribution. We use the basic forces q as boundary values ofthe problem to obtain the statement of equilibrium:

(6.45)

The matrix represents the force-interpolation functions and can be regarded also as an equilibriumtransformation matrix between section forces and basic forces q. In the presence of element loads,the internal forces represent the particular solution of the differential equations in 6.44, which only needto satisfy homogeneous boundary conditions. For uniform element loads the axial force is a linearfunction and the bending moment is a quadratic function. Denoting the particular solution by ,the equilibrium equations are

(6.46)

After setting up the equilibrium relations the geometric compatibility of the frame element can beestablished with the principle of virtual forces. The complementary virtual work is analogous to thevirtual work principle in 6.37

(6.47)

Using 6.46 for the equilibrium relation of the virtual force system, , and after substitutioninto 6.47 gives the compatibility statement as

(6.48)

the individual basic deformation quantities in 6.48 can be evaluated with the force-interpolation functions from 6.45:

(6.49)

sq

q q

q

q

q

b q( )( )

( )( )x

N x

M xx

L

x

L

x

L

x

Lx=

ÊËÁ

ˆ¯

= -ÊËÁ

ˆ¯

+

Ê

ËÁÁ

ˆ

¯˜˜

= -ÊËÁ

ˆ¯

È

Î

ÍÍ

˘

˚

˙˙

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜ =

1

2 3

1

2

3

1

1 0 0

0 1

b( )xs( )x

s w( )x

s b q s( ) ( ) ( )x x x= + w

d dq v s eT T= Ú ( ) ( )x x dx

L

d ds b q( ) ( )x x=

v b e= Ú T( ) ( )x x dx

L

b( )x

v

v

v

1

0

2

0

3

0

1

=

= -ÊËÁ

ˆ¯

=

Ú

Ú

Ú

e

k

k

a

L

L

L

x dx

x

Lx dx

x

Lx dx

( )

( )

( )



It is important to emphasize that the compatibility relationships in 6.46, 6.48 and 6.49 hold true for anymaterial response as long as the transverse displacements are sufficiently small that virtual force equilib-rium can be satisfied in the undeformed configuration.

For the special case of linear elastic material response, the section relationship in 6.41 can be invertedto give the section deformations in terms of the section forces. We can generalize it by adding nonme-chanical initial deformations , such as caused by temperature and shrinkage strains

(6.50)

with the section flexibility matrix. Substituting 6.50 into 6.48 and using 6.46 gives

(6.51)

in which f is the element flexibility matrix, are the deformations due to element loads and thedeformations due to nonmechanical effects. In the general case of a tapered frame element with variablecross section, substitution of the force-interpolation functions from 6.45 gives

(6.52)

with x = x/L. For general functions and , 6.52 needs to be integrated numerically. Amongthe various schemes, Gauss or Gauss–Lobatto integration is preferred for the smallest number of functionevaluations for a given accuracy level. Details of these integration schemes can be found in textbooks onfinite element analysis (Bathe, 1995). In the specific case of a uniform prismatic frame element, 6.52simplifies to

(6.53)

For the case of linear elastic material, the deformation–force relation in 6.51 is inverted to give theforce–deformation relation in the following form:

where k is the basic stiffness matrix as the inverse of the flexibility matrix, and and are the fixed-end forces under the element loads and the nonmechanical deformations, respectively. With andf from 6.53, the basic stiffness matrix of a prismatic frame element is

e 0( )x

e k s f s e( ) ( ) ( ) ( ) ( ) ( ) ( )x x x x x x x= [ ] + = +-

s s

1

0 0e

f s( )x

v b f b q s e

v b f b q b f s b e f q v v

= +[ ] +{ }

=È

ÎÍÍ

˘

˚˙˙

+ + = + +

Ú

Ú Ú Ú

Ts w

Ts

Ts w

Tw

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

x x x x x dx

x x x dx x x x dx x x dx

L

L L L

0

0 0

v w v 0

f =-( ) -( )

--( )

--( )

È

Î

ÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙

ÚL

EA

EI EI

EI EI

d

10 0

01 1 1

01

0

1( )

( ) ( )

( ) ( )

xx x

xx xx

x xx

x xx

x

EA( )x EI( )x

f = -

-

È

Î

ÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙

L

EAL

EI

L

EIL

EI

L

EI

0 0

03 6

06 3

q f v v v kv q q= - -( ) = + +-10 0w w

q w q 0

k f= -1



(6.54)

The use of moment releases at the ends of a frame element was presented in Section 6.3.3. For the case

that the element has a moment release at end j and thus , 6.53 shows that , as already

used in the compatibility relation of a prismatic, linear elastic frame element with a moment release in6.20. For the frame element with a moment release at one or both ends, the compatibility relation in6.20 and the contragradient property of force and displacement transformations give

For the case that there is a moment release at end j, with the stiffness matrix k from 6.54 this gives

6.7.2 Concentrated Plasticity Elements

6.7.2.1 Truss or Brace Element

The simplest nonlinear element is the prismatic truss or brace element. In this case there is only onebasic force , which is equal to the normal force in the truss element . The normal force isequal to the axial stress multiplied by the cross-sectional area A. The axial stress is related to the axialstrain by the material constitutive relation. Finally, the axial strain is related to the element deformation

by . Thus, the element state determination is as follows: given , determine ; use thematerial constitutive relation to get the corresponding axial stress ; finally, compute the basic forcewith . The tangent stiffness matrix of the element can be readily obtained by the chain ruleof differentiation

where Et is the tangent modulus of the material.

6.7.2.2 Elastic Perfectly Plastic Beam

The next element of interest is a frame element with concentrated flexural hinges at the ends, wheremoments are assumed to be largest under the combination of gravity and lateral forces due to earthquakeexcitation. Although this is correct for columns, it is often an approximation for girders, where thecombination of gravity and lateral forces, particularly in higher building floors, may lead to the formationof a plastic hinge away from the member ends. In such case, it is advisable to subdivide the member intotwo or more frame elements. The limitation that plastic hinges can only take place at specific locations

k f= =

È

Î

ÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙

-1

0 0

04 2

02 4

EA

LEI

L

EI

LEI

L

EI

L

q3 0= v v3 2

1

2= -

˜ ˜ ˜ ˜ ˜ ˜q a q a kv q q a k a v a q q k v q q= = + +( ) = + +( ) = + +hT

hT

w hT

h hT

w w0 0 0

a k a k ah hT

h=

-

È

Î

ÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙

= =

È

Î

ÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙

1 0 0

0 1 0

01

20

0 0

03

0

0 0 0

˜

EA

LEI

L

q1 s1 = N

v1 e = v1 L v1 es

q s1 1= = sA

kq

v vtt=

∂∂

= ∂∂

∂∂

=1

1 1

AE A

L

se

e



along the span is sufficiently accurate, particularly if one allows for plastic hinges to form at the outerquarter span points using three frame elements for the member.

The simplest way of accounting for the interaction between axial force and bending moment in thepotential plastic hinge locations at the column ends is to use the axial forces from an elastic analysisunder gravity loading to determine the plastic flexural capacity of the hinges. The girders are subjectedto a small axial force so that the variation of this force during the analysis can be neglected. If significantoverturning moments develop in the structure during the nonlinear pushover analysis under lateralforces, the axial force in the columns changes appreciably during the analysis and more sophisticatedmodeling of the axial force-bending moment interaction at the column plastic hinges is required.

The state determination of the frame element can be undertaken with an event-to-event strategy: giventhe element deformations , estimate the basic forces by using as tangent stiffness the elastic stiffnessof the element , i.e., and . The end moments are compared with the correspondingplastic capacities . If these are not exceeded, then the basic force estimate is correct and the tangentstiffness is equal to the elastic stiffness. If the end moments exceed the corresponding plastic capacity,then the ratios and express the amount of overshoot. The first event factor is theinverse of the larger ratio:

The initial deformations are scaled with this ratio, known as an event factor, the moment release codeis set to unity for the end with the event factor, and a new tangent stiffness matrix is formed. Thenew estimate of the basic forces is

q = kt1h1v + kt2(1 – h1)v

If another hinge has not formed, then the basic force estimate is correct and the tangent stiffness of theelement is . If a second hinge forms, then the last deformation increment is scaled by the new eventfactor , the tangent stiffness is updated (which turns out to be zero in the presence of two hinges)and the end forces become

q = kt1h1v + kt2(1 – h1)h2v

With this procedure a maximum of two iterations is required for convergence under monotonic loading.Even under cyclic loading the number of iterations does not exceed two, because the plastic hinges atthe ends can be either open or closed. It is important to note, however, that cyclic loading requires thatthe process be conducted with deformation increments instead of total deformations and that the stateof the hinges, the basic forces and the tangent stiffness matrix be saved from one iteration of the globalequilibrium equations to the next.

6.7.2.3 Two-Component Parallel Model

The frame element with concentrated plastic hinges at the ends is straightforward in its implementation,but is also limited to elastic-perfectly plastic behavior. Thus, it can be a useful addition to the plasticanalysis of Section 6.2.3 by providing the complete force–displacement relation of the structural modelup to incipient collapse. For a more realistic representation of material behavior the inclusion of strainhardening is important. In such case the simple elasto-plastic frame element needs to be combined inparallel with a linear elastic frame element to form the two-component model (Clough et al., 1965).

The two-component model consists of an elasto-plastic element in parallel with a linear elastic element.The latter represents the strain hardening response of the frame member. The fact that the elements arein parallel implies that

(6.55)

vk e k kt1 e= q k v= t1

q p

q q2 2p q q3 p3 h1

h1

2

2 3

=Ê

ËÁˆ

¯min ,

q

q

q

qp p3

k t2

k t2

h2

v v v q q q k k k= = = + = +e p e p e p



where subscripts e and p denote the elastic component and the elasto-plastic component, respectively.The state determination process of the element is straightforward, because of the first relation in 6.55:given v the deformations of each component are known. For the elastic component the basic forces are

and for the elasto-plastic component the state determination process of the preceding sectiondetermines the end forces. The same is true for the stiffness. Once the end forces and tangent stiffnessof the elasto-plastic element are determined, then the last two equations in 6.55 yield the resisting forcesand the stiffness of the entire element. This simplicity of the state determination process is characteristicof elements with an assumption about the deformation distribution.

The main difficulty with the two-component model is the calibration of the element parameters. Underuniform curvature the section stiffness of the linear elastic component can be set equal to the strainhardening section stiffness of the frame member. The initial stiffness of the elasto-plastic component canthen be determined to make up the difference between the initial stiffness of the frame member and thestrain hardening stiffness. Unfortunately, the case of uniform curvature is of little use in earthquakeengineering analysis. Rather the calibration of the model parameters takes place under antisymmetriccurvature with the point of inflection at member midspan. Another limitation of the two-componentmodel is its inability to handle different plastic moment capacities under positive and negative curvature.For these reasons the two-component model has been superceded in earthquake engineering analysis bythe one-component model (Giberson, 1967). It is worth discussing in some detail the formulation ofthis element, because it is characteristic of a class of elements that are based on an assumption about theinternal force distribution. These elements play an important role in modern earthquake engineeringanalysis, because they represent exactly the force distribution in the member and result in a robustnumerical implementation.

6.7.2.4 One-Component Series Model

The one-component model consists of a linear elastic element connected in series with a rigid-plasticlinear hardening spring at each end. The conditions governing the response of the element are

(6.56)

It is convenient to establish these relations with a substructure approach to the statically determinatestructure. Limiting attention to the flexural contribution, the force-interpolation functions for the basicforces and are

and the composite flexibility matrix of the subelements is

where and are the spring flexibilities at ends i and j, respectively. From the product weobtain the last relation in 6.56 with given by the flexural terms in 6.53 and by the followingexpression

q k ve e e=

q q q v v v f f f= = = + = +e p e p e p

q 2 q3

b =È

ÎÍ

˘

˚˙

1 1 0 0

0 0 1 1

f

f

f

c

i

j

=-

-

È

Î

ÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙

0 0 0

03 6

0

06 3

0

0 0 0

L

EI

L

EIL

EI

L

EI

f i f j b f bTc

f e f p

ff

fp

i

j

0

0=

È

ÎÍÍ

˘

˚˙˙



The flexibility coefficients and are zero for a moment less than the flexural plastic capacity andthen assume a value equal to the inverse of the strain hardening stiffness. Because the two end springsare independent, their properties can also be specified independently. Moreover, it is possible to specifya different plastic capacity and a different strain hardening stiffness under a positive than under a negativecurvature. This makes the one-component model more versatile than the two-component model. Anotheradvantage of this model is its ability to account for the effect of element loads by the inclusion of from 6.51 so that with f from 6.56. The calibration of the model parameters takes placeunder antisymmetric curvature with the point of inflection at member midspan. This may not be areasonable approximation for the case of a different plastic capacity under a positive than under a negativecurvature.

It is important to discuss the process of element state determination for the one-component model,because it is characteristic of the class of elements that are based on an assumption about the internalforce distribution. Because the element is implemented in a standard computer program that is basedon the direct stiffness method of analysis, it is expected to return the resisting forces and current stiffnessmatrix for given element deformations v. In order to highlight the fact that these deformations are givenwe denote them with the symbol . From the middle equation in 6.56 we have

Because the deformations of the elastic and plastic components of the one-component model are func-tions of the basic forces q we formally write that

(6.57)

and note that we are dealing with a nonlinear system of equations, because of the rigid-plastic linearhardening component. The solution of the nonlinear system can be obtained with the Newton–Raphsonalgorithm, as will be discussed in more detail in the following section. We defer, therefore, the discussionof the state determination process of this class of elements until then. We note at this stage, however,that 6.57 implies an iterative process of state determination at the element level. An alternative approachthat bypasses the element iteration is also possible.

6.7.3 Distributed Inelasticity Elements

The limitation of concentrated plasticity elements is that inelastic deformations take place at predeter-mined locations at the ends of the element. While this may be a reasonable assumption in lower floorsof moment-resisting frames, it does not account for the possibility of inelastic deformations taking placewithin the element in the upper floors of the building model. Another, in many respects more seriouslimitation, is the fact that concentrated plasticity elements require calibration of their parameters againstthe response of an actual or ideal frame element under idealized loading conditions. This is necessary,because the response of concentrated plasticity elements derives from the moment–rotation relation oftheir components. In an actual frame element the end moment–rotation relation results from theintegration of the section response. This can be achieved directly with elements of distributed inelasticity.In this case there are two approaches: the force formulation or the displacement formulation.

In the force formulation we make use of the fact that the internal forces at a distance x from endi of a two-node frame element are given as the product of the force-interpolation functions andthe basic forces q according to 6.45. In the presence of element loads we need to modify this relationaccording to 6.46. It is important to note that these relations hold for any material response, as long asthe equilibrium can be satisfied in the undeformed configuration. The element deformations can thenbe established by the principle of virtual forces from 6.48. This implies that the section deformations

can be obtained from the section forces . In reality, this relation is not available, but its inverseis. Thus, needs to be established from the solution of the nonlinear system of equations

f i f j

v w

v fq v= + w

)v

)v v v v v- = = +0 where e p

)v v q- =( ) 0

s( )xb( )x

e( )x s( )xe( )x



For the solution of the nonlinear system of equations in 6.57 we also need to establish the change of theelement deformations with q. This change is reflected by the derivative of the expression in 6.48 withrespect to q. We obtain

(6.58)

where is the section flexibility, which can be obtained as the inverse of the section stiffness matrixin 6.39. The derivative of the section forces with respect to q is obtained from 6.46. We call the expressionin 6.58 the tangent flexibility matrix of the element.

In the displacement formulation we assume that the axial and transverse displacements at distance xfrom the end node i are supplied by the product of suitable displacement interpolation functions with the element deformations v. For a two-dimensional element we write

The displacement interpolation functions correspond to the solution of the differential equation for alinear elastic, prismatic frame element. It is important to note that these functions thus only approximatethe response of a frame element with distributed inelasticity. This has important ramifications underlarge inelastic deformations, as a later example will demonstrate. The section deformations at x can beobtained by application of the small deformation theory of beam kinematics: the axial strain at thereference axis is the first derivative of the axial displacement, and the curvature is the second derivativeof the transverse displacement. We write formally

(6.59)

The element forces q are established from the principle of virtual displacements in 6.37 using the variationof 6.59 for the virtual displacement field. We obtain

(6.60)

The section forces in 6.60 are directly obtained from the section constitutive relation for givensection deformations . The latter, in turn, are directly obtained from the given element deformationsv by 6.59. Thus, the path from the given element deformations v to the corresponding forces q isstraightforward in the displacement formulation, which explains its appeal. This should not distract fromthe serious drawback of the method, which lies in the approximate nature of the displacement interpo-lation functions.

The change of element forces q with deformation is also required. This is obtained from the derivativeof 6.60 with respect to v, which yields

b q s s e( ) ( ) ( )x x x+ - ( ) =w 0

∂∂

= ∂∂

= ∂∂

∂∂

=Ú Ú Úv

q qb e b

e

s

s

qb f bT T T

s( ) ( ) ( )( )

( )

( )( ) ( ) ( )x x dx x

x

x

xdx x x x dx

L L L

f s( )x

f t

a( )x

u a va

a a

v

v

v

( ) ( )( )

( ) ( )x x

x

x x= =

È

ÎÍ

˘

˚˙

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜

1

2 3

1

2

3

0 0

0

e a

eu

u

a

a a

v

v

v

ae( )( )

( )

( )

( ) ( )x x

x

x

x

x

x

x

x x=

∂∂

∂∂

È

Î

ÍÍÍÍ

˘

˚

˙˙˙˙

Ê

ËÁÁ

ˆ

¯˜˜

=

∂∂

∂∂

È

Î

ÍÍÍÍ

˘

˚

˙˙˙˙

È

Î

ÍÍ

˘

˚

˙˙

Ê

Ë

ÁÁÁÁÁ

ˆ

¯

˜˜˜˜

=0

0

0

0

0 0

02

2

1

2

2

2

1

2 3

1

2

3

(( )x v

q a s= Ú eT( ) ( )x x dx

L

s( )xe( )x



(6.61)

where is the section stiffness matrix from 6.39, and the derivative of the section deformationswith respect to v is obtained from 6.59. The expression in 6.61 is the tangent stiffness matrix of theelement.

The expressions in 6.48, 6.58, 6.60 and 6.61 for the state determination of the distributed inelasticityelements involve integrals over the element length. The evaluation of these integrals is accomplishednumerically. In analogy with 6.42 the integrals are evaluated as

The most suitable numerical integration methods use the Gauss or the Gauss–Lobatto rule whichoptimize accuracy of smooth integrands for a given number of integration points (Bathe, 1995). TheGauss–Lobatto rule is particularly suitable when it is important to include the ends of the element inthe evaluation. This is indeed the case in earthquake engineering applications, where the largest inelasticdeformations quite often take place at the element ends. Four integration points suffice for the integralsin 6.48, 6.58, 6.60 and 6.61 as long as we are not interested in the effect of the midspan section. In thelatter case, which is important in girders under significant gravity loads, five integration points arerecommended. However, the inclusion of the effect of gravity loads can only be accommodated with theforce formulation. By contrast, in the displacement formulation, the section forces in 6.60 are derivedfrom the section deformations and do not include the effect of loads on the elements, such as due togravity. The latter are included as external virtual work contribution by modification of 6.60 according to

(6.62)

We conclude from 6.62 that the initial or fixed-end forces of the displacement formulation are notdifferent from those of a linear elastic, prismatic frame element. Moreover, in this case the element loadsonly affect the element forces and do not affect the internal force distribution. This is another seriouslimitation of the displacement formulation.

6.8 Solution of Equilibrium Equations

The equilibrium equations in 6.1 constitute the starting point for linear or nonlinear structural analysismethods. We focus our attention on the free DOFs of the model and write for the applied and resistingforces the system of equations:

(6.63)

The subscript f is dropped for brevity of notation, and the reactions at the restrained DOFs can beevaluated after the equations for the free DOFs are solved. In the general case the resisting forces areimplicit functions of the displacements U at the DOFs of the structural model and 6.63 becomes a setof nonlinear equations in the unknown displacements U

(6.64)

∂∂

= ∂∂

= ∂∂

∂∂

=Ú Ú Úq

v va s a

s

e

e

va k ae

Te

Te

Ts e( ) ( ) ( )

( )

( )

( )( ) ( ) ( )x x dx x

x

x

xdx x x x dx

L L L

k s( )xk t

g x dx w g x

L

l l

l

nIP

( ) ( )Ú Âª=1

q a w a s+ =Ú ÚTe

T( ) ( ) ( ) ( )x x dx x x dx

L L

P Pf rf- = 0

P P U- =r( ) 0



where we assume that the applied forces P do not depend on the displacements U. Equation 6.64 appliesto static analysis. For dynamic analysis it will be generalized with the inclusion of inertia effects in a latersection.

6.8.1 Newton–Raphson Iteration

To develop a solution algorithm for the nonlinear system of equations in 6.64, the resisting forces areexpanded in a Taylor series about an initial displacement vector :

(6.65)

Truncating the Taylor series after the linear term and substituting the expression of the resisting forcesfrom 6.65 in 6.64 we obtain a linear system of equations for the unknown displacements U. Denotingthe displacement increment by , the linearized equilibrium relationship from 6.65 is

(6.67)

Equation 6.67 includes the tangent stiffness matrix, , for the free DOFs of the structural model asthe derivative of the global resisting force vector with respect to the displacements. This derivative isevaluated at U0 in 6.67. The term (l, m) of the stiffness matrix represents the partial derivative of theresisting force at DOF l with respect to the displacement at DOF m. The tangent stiffness matrix isobtained by direct assembly of the tangent stiffness matrices of the elements in the structural model afterthe latter have been transformed to the global coordinate system, as will be discussed later in this section.

The solution of 6.67 yields the displacement increment . An improved estimate of the solutionto the system of equilibrium equations in 6.64 can be obtained with . The repetition ofthis process will converge to the solution of the nonlinear set of equilibrium equations in 6.64 undercertain conditions. This iterative procedure is known as Newton–Raphson algorithm. An importantcharacteristic of the Newton-Raphson iteration is that the process converges with a quadratic rate to thesolution. This can be observed by a slight modification of 6.65:

where the last equation results from the use of 6.67 for the first two terms of the second equation. Bytaking absolute values of the last expression we arrive at the desired result , wherec is a constant. This means that the error between the solution and iterate is less than the square of theerror for the previous iterate. This characteristic is used in simulation studies to ascertain that the stiffnessmatrix is indeed tangent to the structural response, because the latter fact can have important ramifica-tions for the convergence of the Newton–Raphson algorithm. The constant c depends on the secondderivative of the resisting force, or the change in the tangent stiffness. Large changes in stiffness result

U 0

P U P UP

UU U

P

UU Ur r

r0

r0( ) ( ) ...= +

∂∂

-( ) +∂∂

-( ) +00

2

2

0

21

2

DU U U= - 0

P P UP

UU K U- =

∂∂

=rr

t0( )00

D D

K t

DUU U U1 0= + D

P U P UP

UU U U U

P

UU U

P U P UP

UU U

P

UU U

P

UU U

P

UU U

r rr

0r

0

r rr

0r r

0

r

( ) ( )

( ) ( )

= +∂∂

- + -( ) +∂∂

-( )

= +∂∂

-( ) +∂∂

-( ) +∂∂

-( )

=∂∂

-

00

1 1

2

2

2

00

10

1

2

2

2

0

1

2

1

2

0

x

x

11

2

2

21

2( ) +

∂∂

-( )P

UU Ur

0

x

U U U U- £ -1 0c2



in large constants and slower convergence. As will be covered in the following, the convergence can beimproved by a load incrementation strategy.

The Newton–Raphson algorithm proceeds in the following steps:

1. At the start of iteration j, compute the resisting forces and the tangent stiffness for the displacements at the end of the previous iteration .

2. Solve the linearized system of equations for .3. Update the displacement vector , advance the iteration index and return to step 1

repeating steps 1 to 3 until convergence. The convergence criterion will be discussed later in this section.

The convergence of the algorithm depends on the initial displacement to start the iteration, the charac-teristics of the tangent stiffness matrix and the convergence criterion. It is important that the initialguess be close to the actual solution.

6.8.2 Load Incrementation

To improve the convergence of the Newton–Raphson algorithm for nonlinear structural analysis, it isnecessary to incorporate a procedure for incrementing the load to limit the changes in the structuralstate for each load increment. Instead of applying the load in one step, the solution is divided into severalsteps, and it proceeds with load factor increments whose magnitude can be adjusted depending on thestate of the structure. Within each load step the Newton–Raphson iteration process can be used to solvethe equilibrium equations.

We assume that the applied loads are grouped in load patterns with independent histories. The simplestcase is a single applied force pattern , and this case has some important applications in the nonlinearstatic analysis of structures under equivalent lateral earthquake loads. The notation for the load incre-mentation procedure uses a superscript in parentheses to denote the load step number with index k whilea subscript denotes the iteration number in each load step with index j. The applied force vector at loadstep k can, therefore, be written as

where the load factor results from a series of increments,

so that the applied force vector at load step k can also be written as

The equilibrium equations at load step k become

and the starting displacement vector for satisfying the above equilibrium equations is , thatis the state at the end of the previous load step. If the load factor increment is held constant during theequilibrium iterations, then the iteration process consists of the three steps presented earlier with theunbalanced force given by . Only the resisting force vector is updatedduring the equilibrium iterations.

6.8.3 Load Factor Control during Incrementation

The load factor increment can be changed in the Newton–Raphson algorithm so that large incrementsare applied when the structure stiffness does not change much, and the increments are reduced when

P P Ur rj j- -= ( )1 1

K tj-1 U j-1

P P P K Uu r tj j j j= - =- -1 1D DU j

U U Uj j j= +-1 D

K t

Pref

P P( ) ( )k k= l ref

l( )k

l l l( ) ( ) ( )k k k= +-1 D

P P P( ) ( ) ( )k k k= +-1 Dl ref

P P P P P( ) ( ) ( ) ( ) ( )k k k k k- = + --r ref r

1 Dl

U U0( ) ( )k k= -1

P P P Pu ref rjk k k

jk( ) ( ) ( ) ( )= + --

-1

1Dl



the stiffness changes. An excellent parameter for this purpose is the current stiffness parameter (Bergan,1978). This parameter is the scalar product of the reference force vector and the corresponding displace-ments caused by the forces. Denoting the displacements under the reference force vector with , thestiffness parameter is defined as

The initial value of the parameter is computed with the initial tangent stiffness matrix. Thefollowing expression gives the change in the load factor for load step k (Bergan, 1978),

(6.68)

where is the stiffness parameter at the end of the previous load step k-1, is the load factorincrement for the first load step with k=1, commonly selected to be about 30% of the collapse load factor,and is a constant between 0.5 and 1.2, with 1.0 being a commonly used value. Figure 6.19 shows theload–displacement response of a structural model with an exponent value of . The model is a trussstructure with nonlinear material response. It is clear from the figure that with load factor adjustmentduring incrementation it is possible to approach the maximum strength of the model even though thetangent stiffness becomes nearly singular.

6.8.4 Load Factor Control during Iteration

Load incrementation allows the Newton-Raphson algorithm to approach the maximum strength, but itcannot compute the postpeak response. For tracing the postpeak response, the incrementation procedureneeds to be further modified so as to control the load during the equilibrium iterations. In this case theapplied force vector is updated during the equilibrium iterations of a load step. With a subscript denotingthe iteration number, the unbalanced force is

(6.69)

FIGURE 6.19 Load–displacement response with load factor control during incrementation.

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

Vertical Displacement

0.6

0.8

1

1.2

1.4

1.6

Load

fact

or λ

U t

S p = = -P U P K PrefT

t refT

t ref1

S p( )0

D Dl l

g

( ) ( )

( )

( )k p

pk

S

S= -

0

0

1

S pk( )-1 Dl( )0

gg = 1

P P P P P Pu r ref rj

kj

kj

kjk

jk

jk( ) ( )

-( ) ( ) ( )

-( )= - = + --1 1 1Dl



During the first iteration, , the first term on the right-hand side of 6.69 is equal to the force vectorat the end of the previous load step, . Since all superscripts in 6.69 refer to load step k, theseare not included in the following equations for brevity of notation. Using the Newton-Raphson algorithm,the following system of linear equations of equilibrium is solved at each iteration:

(6.70)

Substituting 6.69 into 6.70 gives

(6.71)

so that

With the decomposition of displacement increments into two terms on the right-hand side of 6.71, aseparate condition can be introduced to determine the load factor increment during iteration j. Severalschemes have been proposed for the purpose and there is extensive literature on the subject (Clarke andHancock, 1990, Crisfield, 1991). The schemes that prove particularly useful for the nonlinear static(pushover) analysis of structures with equivalent lateral earthquake loads involve load control duringiterations under constant displacement at a single selected degree of freedom (sometimes referred to asthe control node). In this case, we impose a condition on degree of freedom n, such that

, to determine the load factor increment . An alternative is to usethe condition of constant external work to determine the load factor, which leads to the condition that

. After substituting from 6.71, the load factor is given by

With load control during equilibrium iterations the steps of the Newton–Raphson algorithm for a singleload step are:

1. Start with the structure state determination for the displacements at the end of the previousiteration in load step k and determine the tangent stiffness matrix and resisting forcevector . The applied force vector is also known since .

2. Compute the displacements under the reference force vector and the residual displacementincrements with the equations following 6.71. Since this step involves the solution of asystem of equations under different force vectors, the tangent stiffness need only be factored oncefollowed by separate back substitutions for each system of equations.

3. Determine the load factor increment and the resulting displacement increments :

4. Update the displacements and the load factor:

j = 1P P0

1( ) ( )k k= -

P K Uu tj j j= -1D

P P P K U K U Uj j j j j j j j j- - -+ - = = +( )1 1 1 1 1 1D D D Dl lref r t t r t- - -

P P K U

P K U

j j j j

j j

- - =

=

1 1 1 1

1 1

r t r

ref t t

- - -

- -

D

D D DU U Uj j j jn tn rn= + =- -l

1 10 Dl j

D D DWj j j= =U PTrefl 0 DU j

DD

l j

j

j

= - -

-

U P

U P

rT

ref

tT

ref

1

1

j - 1 K t j-1

Pr j-1Pj-1 P Pj j- -=1 1l ref

U tj-1

DU rj-1

Dl j DU j

D D DU U Uj j j j= +- -l t r1 1

U U Uj j j

j j j

= +

= +

-

-

1

1

D

Dl l l



Update the iteration index and repeat steps 1 through 4 until convergence. Convergence is measured bythe ratio of the relative work increment in iteration j to the initial work increment at iteration

where

Figure 6.20 shows the load–displacement response of the same truss structure as in Figure 6.19, for thecase that the truss elements have a softening modulus equal to 10% of the initial modulus. The figureshows how the load control algorithm permits the tracing of the postpeak response.

6.8.5 Structure State Determination

A key step in the Newton–Raphson algorithm is the structure state determination, which involves thedetermination of the resisting force vector and structure stiffness matrix for given structural displace-ments and their increments. The tangent stiffness matrix of the structure is obtained by the partialderivative of the resisting forces with respect to the displacements at the global DOFs. Applying the chainrule of differentiation to 6.1 gives

(6.72)

The assembly operation on the right-hand side involves row indexing and summation of the elementcontributions. Each row of the element resisting forces will produce as many columns as the number of

element DOFs in vector u in the operation . With 6.12 these columns will be postmultiplied

by a matrix with a single nonzero term of unity in each row. This unit value is located at the columnthat corresponds to the global DOF number to which the element DOF in the corresponding row maps.

Postmultiplication by this matrix amounts to mapping the column numbers of to the column

numbers corresponding to the global DOF numbers for the DOFs of this element. From 6.72 we conclude

FIGURE 6.20 Load–displacement response with load factor control during iteration.

0 0.02 0.04 0.06

Displacement

0.08 0.1 0.120

0.1

0.2

0.3

0.4

0.5Lo

ad fa

ctor

λ 0.6

0.7

0.8

0.9

DWj

j = 1

D D DWj j j j j= + -( )- -U P P PT

ref r1 1l

KP

U UP

p

u

u

UAt

rr,=

∂∂

= ∂∂

= ∂∂

∂∂Â id

el

elel

el

el

el( )

( )

( )

( )

A p

uel

el

el

∂∂

( )

( )

∂∂

p

u

( )

( )

el

el



that the global tangent stiffness matrix of the structure can be obtained by direct assembly (row

indexing and summation) of the element stiffness coefficients in the global coordinate system, as longas the columns of the element stiffness matrix are mapped to the global DOF numbers correspondingto the element DOFs of the particular element. This is written in compact form as

(6.73)

The assembly of the resisting force vector follows 6.1.The partial derivative of the element forces with respect to the element displacements in the global

reference system represents the element tangent stiffness matrix in the global coordinate system. Thetangent stiffness matrix of the element can be obtained from 6.5 by the chain rule of differentiation,

(6.74)

where is the tangent stiffness of the basic element. Linear geometry is assumed in 6.74 so that thetransformation matrix does not depend on the displacements u, and 6.18 can be used for the derivativeof the element deformations with respect to u.

6.8.6 State Determination of Elements with Force Formulation

The Newton–Raphson algorithm can be used for the solution of the system of nonlinear equations in6.57. We present it here because the state determination of this class of elements is not well known inthe literature. For iteration j

Noting that the element stiffness is the inverse of the flexibility matrix, we can obtain the increment ofthe element forces according to

and update the element forces to . We solve for the section deformations and incrementsat the integration points from

noting that the derivative of the section forces with respect to section deformations is the last sectionstiffness matrix . After determining De j(x), we update the section deformations to

. Finally, we determine the current element deformations from the principle ofvirtual forces

and return to the beginning of the algorithm with a new deformation residual .

K t

K kAt t=el

el( )

Pr

kp

u ua q a

q

v

v

ua k at g

TgT

gT

t g( )el = ∂

∂= ∂∂ ( ) = ∂

∂∂∂

=

k t

ag

)v v

v

qq- - ∂

∂=-

-j

j

j1

1

0D

Dq j

Dq k v vj j j= -( )- -t 1 1

)

q q qj j j= +-1 D

b q s ss

ee( ) ( ) ( ) ( )x x x xj j

jjj

+ - + ∂∂

Ê

ËÁ

ˆ

¯˜ =-

-w 1

1

0D

k s jx-1

( )e e ej j jx x x( ) ( ) ( )= +-1 D

v b ej j

L

x x dx= Ú T( ) ( )

)v v- j



The above process implies an iterative element state determination for each iteration of the solutionof the global equilibrium equations (Ciampi and Carlesimo, 1986, Taucer et al., 1991). An alternativescheme has also been used with success whereby the element state determination consists of a singleiteration that works in tandem with the global equilibrium equations (Neuenhofer and Filippou, 1997).In either case by setting in the element state determination algorithm we conclude that we needto store the element forces and the section deformations from a global iteration to the next.

6.8.7 Nonlinear Solution of Section State Determination

The section state determination forms part of the algorithm of the state determination of elements withdistributed inelasticity, but is also important in its own right in the determination of moment–curvatureand interaction diagrams of sections. Therefore, we briefly describe the process of section state determi-nation for a couple of cases. In the simplest case the section deformations e are given. Using theassumption of section kinematics we determine the strain at the integration points of the cross section

according to the first equation in 6.43 and use the material constitutive relation to obtain thecorresponding stress and material stiffness. We then determine the section forces s and the section stiffness

by numerical evaluation of the integrals in 6.43. This is the process used at every integration pointof the distributed inelasticity elements during state determination. The same process can be used toobtain the interaction diagram of a cross section, in which case the section deformations e are set so asto describe the limit state of the section.

In the determination of the moment–curvature diagram the problem is slightly different. In this casethe curvatures are specified along with the axial force acting on the cross section. The correspondingaxial strain is determined from the single available equilibrium equation, the axial force equilibrium. Wewrite symbolically

where the symbol over the axial force N denotes the given value. Because this is a scalar equation, differentsolution methods can be used. We show here the application of the Newton–Raphson solution. For thiswe write for the step of load incrementation

where denotes the axial force value of the last load step. We note also that the derivatives of the axialforce with respect to the section deformations correspond to terms (1,1) and (1,2) of the section stiffnessmatrix . Because the curvature increment is specified, we solve the above equation for the initialincrement of the axial strain according to

During subsequent iteration corrections the curvature increment is set equal to zero.

6.9 Nonlinear Geometry and P-DDDD Geometric Stiffness

The element tangent stiffness matrix in Section 6.8.5 was derived on the assumption of linear geometry,in which case the element equilibrium equations are satisfied in the undeformed configuration and the

j = 1q j-1 e j x-1( )

el

k s

)N N a- ( ) =e k, 0

)N N

N N

aa- + ∂

∂+ ∂∂

Ê

ËÁˆ

¯=0 0

ee

kkD D

N 0

k s

De a

∂∂

= - + ∂∂

ÊËÁ

ˆ¯

NN N

N

aae

ek

kD D)

0



compatibility relation between element deformations and end displacements in the global referencesystem does not depend on the displacements. In the general case of nonlinear geometry, the elementequilibrium equations need to be satisfied in the deformed configuration according to Section 6.4, andthe compatibility relation between element deformations and end displacements in the global referencesystem becomes nonlinear on account of large displacements. For applications in earthquake engineering,approximations of the nonlinear equilibrium and compatibility relations are possible, as will be discussedin the following.

6.9.1 Resisting Forces and Element Tangent Stiffness Matrix

As discussed in Section 6.4, the equilibrium in the deformed configuration is expressed in the localcoordinate system. Thus, the resisting forces for given end displacements in the local coordinatesystem are given by either 6.21, 6.22 or 6.23 depending on the desired accuracy in the geometricallynonlinear analysis. The compatibility transformation is used to transform the end displacementsfrom the global to the local coordinate system. After obtaining the element end forces in the localcoordinate system from either 6.21, 6.22 or 6.23, the rotation transformation transforms theelement resisting forces to the global coordinate system.

For the element stiffness matrix, we proceed in an analogous manner:

(6.75)

There is nothing special in this transformation process relative to linear geometry. However, the tangentelement stiffness matrix in the local reference system requires additional consideration for geomet-rically nonlinear analysis. Using the resisting forces in 6.21, the chain rule of differentiation for theelement stiffness matrix in the local reference system gives

(6.76)

The element stiffness matrix in the local coordinate system in 6.76 is composed of two contributions:

the geometric stiffness arising from the change of the equilibrium matrix with end displacements

, and the material stiffness , which represents the transformation of the tangent basic stiffness

to the local coordinate system.

Using the deformation-large displacement relations in 6.13 and 6.14, the derivative of v relative to is as follows:

By comparison of the above expressions with 6.21, it is clear that contragradience holds, so that

p u

u a u= r

pp a p= r

T

∂∂

= ∂∂ ( ) = ∂

∂∂∂

=p

u ua p a

p

u

u

ua k ar

TrT

rT

r

k

k

kp

u ub q

b

uq b

q

v

v

u= ∂∂

= ∂∂

( ) =∂∂

+ ∂∂

∂∂u

uu

k g

u k m

kq

vt = ∂∂

u

∂∂

=∂∂

= - +( ) - +( )∂∂

= ( ) - -+

-+Ê

ËÁˆ

¯

∂∂

= ( ) -

v

u uu u u u

v

u

u u u u

v

u

u

1

2

3

10 0

0 0 1 0 0 01

0 0

0 0 0 0 0 11

L

L

L L

L LL L

L L

L

L L

L

L

L

n n n

nx y x y

n

y

n

x

n

y

n

x

n

n

D D D D

D D D D

D yy

n

x

n

y

n

x

nL

L

L L

L

L-

+-

+Ê

ËÁˆ

¯D D Du u u

0 0

∂∂

= =v

ua bu u

T



when the difference between and L in is neglected. Thus, 6.76 becomes

(6.77)

The form of 6.77 reinforces the earlier statement that the material stiffness is equal to the tangent basicstiffness transformed to the local coordinate system. Combining 6.77 with 6.75 gives the element stiffnessmatrix in the global coordinate system including the geometric and material stiffness contributions:

(6.78)

It is important to emphasize that 6.78 holds for all element types, as long as the element force-deformationrelation is defined in the basic system. Of equal importance is the fact that by separating the geometrictransformations from the actual element formulation in 6.78 different geometric theories can be imple-mented for the same element by selecting the form of the compatibility matrix . Under linear geometrymatrix simplifies to 6.17 and is displacement independent. Thus, the geometric stiffness is zero andthe element stiffness matrix reduces to 6.74 with , as defined in 6.18 without rigid end offsets.

6.9.2 Geometric Stiffness Matrix

There remains the task of determining the geometric stiffness in 6.77 and 6.78 by taking the derivative. This operation is performed column by column noting that the derivative of the first column

multiplies the axial force and supplies the contribution to the geometric stiffness matrix, whereasthe derivatives of the second and third columns multiply the end moments and , respectively.These can be combined to give the contribution to the geometric stiffness matrix. From thederivative of the first column of the contribution is

(6.79)

in which

From the second and third columns of we obtain

(6.80)

Ln ∂ ∂v u1

k k kb

uq b k b

a

uq a k a k

q

v= + =

∂∂

+ =∂∂

+ = ∂∂g m

uu t u

T uT

uT

t u twith

k aa

uq a k a a a k a a a k a ael

rT u

T

uT

t u r rT

g r rT

uT

t u r=∂∂

+Ê

ËÁˆ

¯= +

au

au

a aag r=

k g

∂ ∂b uu

q1 k g1

q 2 q3

k g23

bu k g1

kq

g1 =

- -- -

- -- -

È

Î

ÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙

1

2 2

2 2

2 2

2 2

0 0

0 0

0 0 0 0 0 0

0 0

0 0

0 0 0 0 0 0

L

s cs s cs

cs c cs c

s cs s cs

cs c cs cn

cL

Ls

Lx

n

y

n

=+

=D Du u

and

bu

kq q

g23 =+

- - - +- - + -

- + - -- + - -

È

Î

ÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙

2 3

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

1

2 0 2 0

2 0 2 0

0 0 0 0 0 0

2 0 2 0

2 0 2 0

0 0 0 0 0 0

L L

sc c s sc c s

c s cs c s cs

sc c s sc c s

c s cs c s csn n



In the scalar factor for the matrix in 6.80, the deformed element length is used in the denominatorbecause the first term represents the shear force in the deformed element configuration. With thisidentification a comparison of the two matrices in 6.79 and 6.80 is possible. In slender compressionelements the shear force is often significantly less than the axial force, so that the contribution isin such cases significantly smaller than and can often be neglected.

6.9.3 P-DDDD Geometric Stiffness

We introduce now an approximation of nonlinear geometry that is often used in structural analysis forearthquake-resistant design. It is known by the name analysis, but this name is confusing for thefollowing reason: when referring to a single member it is convenient to distinguish between the effect ofaxial force on the free body equilibrium of the entire member, the so-called effect and the effectof the axial force on the internal forces in the deformed configuration, the so-called effect. Suchdistinction is, however, not relevant when the member is subdivided into several elements. In such case,even an element that only accounts for the effect can approximate the effect of the axial force onthe internal forces of the member. The greater the number of elements used to model a compressionmember the more accurate the approximation of the internal forces.

To avoid confusion, we refer to the approximation of the exact geometric transformation by the name truss geometric stiffness. In this case we assume that the equilibrium matrix is given by in

6.23 and the resulting geometric stiffness matrix is

(6.81)

Equations 6.81 and 6.79 are similar, leading to the observation that the former can be obtained from thelatter by setting , and . A difficulty arises, however, in using a consistent deforma-tion–displacement relation for . We do not pursue this subject further here and note that com-monly is set equal to the linear compatibility matrix a in 6.77. With these assumptions, the elementstiffness in the global coordinate system is

(6.82)

Although simple, the element stiffness matrix in 6.82 deviates quickly from the tangent stiffness to theelement-force deformation relation, thus resulting in poor convergence behavior even under moderatedisplacements.

The geometrically nonlinear behavior with the exact transformation in 6.77 and the matrices in 6.79and 6.80, or, with the approximate geometrically nonlinear behavior in 6.82 can be improved as necessaryby subdividing the compressed member into smaller elements. In such case, the deformations of theactual member relative to the chord of the elements can be made as small as necessary, so that even avery simple basic force–deformation relation can yield excellent results. Thus, the trade-off is between asmaller number of elements with more accurate force–deformation relation and a large number of verysimple elements. The decision as to which approach to follow depends on the element library of thecomputer software used for the structural analysis. The power of the corotational approach lies in itsability to represent accurately nonlinear geometry under large displacements with simple basicforce–deformation relations. We will illustrate this in the examples of the following section.

k g23

k g1

P - D

P - DP - d

P - D

P - D bPD

k ku

b qq

g P P= = ∂∂

ÊËÁ

ˆ¯

=

-

-

È

Î

ÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙

D D1

0 0 0 0 0 0

0 1 0 0 1 0

0 0 0 0 0 0

0 0 0 0 0 0

0 1 0 0 1 0

0 0 0 0 0 0

L

c ª 1 s ª 0 L Ln ª∂ ∂v u

au

k a k a a k aelrT

P r gT

t g= +D



6.10 Examples of Nonlinear Static Analysis

The examples in this section provide a brief overview of the type of problems that are encountered instructural analysis for earthquake engineering.

The response of sections made of one or several materials is of interest in the assessment of localresponse. Furthermore, it is an important ingredient in the determination of hysteretic element response.The selection of the number of integration points for sufficient accuracy is of particular interest. Figure6.21 shows the axial force-bending moment interaction diagram of a rectangular section with bilinearelastic-perfectly plastic material. Because the results are normalized with respect to the plastic axial andflexural capacity, and , respectively, the dimensions and material properties are not relevant.Figure 6.21 shows clearly that the numerical solution with eight (8) midpoint evaluations for the integralsof 6.43 gives excellent accuracy. The same is true for the monotonic moment–curvature relation of thesame section under three different axial load levels of 0, 30 and 60% of the plastic axial capacity inFigure 6.22. In this example the closed-form solution is available and the comparison shows that a fewmidpoint evaluations suffice for the accurate representation of the section response. The section stiffnessin 6.43 involves quadratic terms of the coordinates and is, therefore, more sensitive to the number ofintegration points. Usually 10 to 15 midpoint evaluations suffice for the purpose. Under biaxial loading25 (5 ¥ 5) to 64 (8 ¥ 8) fibers yield excellent accuracy. For wide flange sections three layers in each flangeand four layers in the web are recommended. In a reinforced concrete section the hysteretic response isdominated by the behavior of reinforcing steel. Thus, it is important to represent the area and distributionof reinforcement relatively well and then use 16 (4 ¥ 4) or 25 (5 ¥ 5) fibers for the concrete. A largernumber of fibers may be necessary for distinguishing between cover concrete and core concrete confinedby transverse reinforcement.

Figure 6.23 shows a three-story steel frame under the action of vertical and horizontal forces. Thematerial is assumed to be elastic-perfectly plastic. The vertical and horizontal forces are collected intothe same reference force vector, so that both are incremented in the following nonlinear static pushoveranalyses. In typical analyses the vertical forces due to gravity are kept constant, while the lateral forcesare incremented to collapse. Figure 6.24 shows the relation between load factor and top story horizontal

FIGURE 6.21 Axial force-bending moment interaction diagram of rectangular section with elastic-perfectly plasticmaterial; analytical and numerical solution with eight midpoint evaluations.

−1−0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Numerical solution

Axi

al F

orce

/Np

Analytical solution

Moment /Mp

N p M p

N p



displacement. The results are obtained with the elastic-perfectly plastic element of Section 6.7.2.2. Figure6.24 shows the load-displacement response for two cases:

• In the first case the plastic flexural capacity does not account for the effect of the axial force• In the second case a linear elastic analysis under the application of the vertical forces yields the

axial forces in the columns. These are used to reduce the plastic flexural capacity of the membersaccording to the LRFD specification. This approach does not account for the change in axial forcein the columns on account of the overturning moments due to the lateral forces. Nonetheless, itgives a relatively accurate estimate of the collapse load factor of the frame, as will be shown later.It is interesting to observe that while the collapse load factor is not very different between the twocases, a collapse mechanism forms at a significantly smaller horizontal displacement in the casein which the effect of the axial force on the plastic flexural capacity is accounted for. The causefor this is apparent from Figures 6.24b and c, which show the collapse mechanism for the two

FIGURE 6.22 Moment–curvature relation for rectangular section with elastic-perfectly plastic material; analyticaland numerical solution with eight midpoint evaluations under axial force of 0%, 30% and 60% of Np.

FIGURE 6.23 Geometry, member sizes and reference loading for three-story, one-bay steel frame.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Numerical SolutionAnalytical Solution

Mom

ent/

Mp

Curvature/κy

50 k

75 k75 k

50 k

25 k25 k

W14 × 159 W14 × 159

W14 × 159W14 × 159

W14 × 159 W14 × 159

200 k

200 k

150 k

13'

13'

13'

15' 15'

W33 × 130

W30 × 116

W27 × 94

75 k 75 k

100 k100 k

100 k 100 k



cases. In the first case an almost complete collapse mechanism forms with nine plastic hinges, asshown in Figure 6.24b. In the second case a partial first story collapse mechanism forms, as shownin Figure 6.24c. Clearly, the difference in collapse mechanism of the two cases significantly affectsthe plastic rotations of the first-story columns for a given horizontal displacement.

Figure 6.25 shows the load factor–top story displacement response of the same frame for the case thata distributed inelasticity element with layer section is used. In this example the force formulation ofSection 6.7.3 is used with five control sections. Each section is discretized into 20 layers, 5 in each flangeand 10 in the web. Figure 6.25 shows the results of two analyses: in the first case the geometry is linear,while in the second the P-D geometric stiffness of Section 6.9.3 is included in the element formulation.

The comparison of the response in Figure 6.25 with that in Figure 6.24a, shows that the axial forcevariation on account of the overturning moments results in a reduction of the collapse load factor. Witha layer discretization of the cross section this effect is automatically accounted for. From the response inFigure 6.25 we conclude that the effect of the P-D geometric stiffness becomes noticeable for values of

(a)

(b) (c)

FIGURE 6.24 (a) Load factor–top story displacement relation for three-story, one-bay steel frame with and withouteffect of axial force on plastic moment capacity (elastic-perfectly plastic frame element). (b) Collapse mechanism ofsteel frame without effect of axial force on plastic moment capacity (magnification factor = 5). (c) Collapse mech-anism of steel frame with effect of axial force on plastic moment capacity (magnification factor =5).

0 0.5

Top Story Horizontal Displacement (ft)

1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

without axial force effectwith axial force effect

Load

Fac

tor l



the top story horizontal displacement larger than 0.35 ft, which amounts to an average story drift of 1%.Clearly, it is essential to include this effect in the pushover analysis of frame structures. Finally, Figure6.25 shows that the load control measures that were discussed in Sections 6.8.3 and 6.8.4 permit thetracing of the load-displacement response past the point of peak strength. This is true for the case ofsoftening response and for the case of linear geometry where the load factor remains practically constantafter attaining the maximum value. Figures 6.25b and c show the distributions of section deformationsand section forces, respectively, at maximum displacement under linear geometry. In the force formula-tion the distribution of section forces is always exact, as reflected by the constant axial force and linearbending moment distributions in Figure 6.25c. The section deformations in Figure 6.25b show that largeinelastic strains take place at the top and bottom end sections of the first-story columns. The accuracyof the inelastic strain estimate depends on the integration weight of the end sections in the elementresponse. In this respect four or five integration points yield results of comparable accuracy to proposalsof plastic hinge length estimation. Figure 6.25b shows that an element with inelastic zones of finite length

(a)

(b) (c)

FIGURE 6.25 (a) Load factor–top story displacement relation for three-story steel frame; distributed inelasticityelement with force formulation and five control sections; layer section with 20 layers. (b) Curvature (solid line) andaxial strain (dashed line) distribution for three-story steel frame at maximum displacement; distributed inelasticityelement with force formulation. (c) Moment (solid line) and axial force (dashed line) distribution for three-storysteel frame at maximum displacement; distributed inelasticity element with force formulation.

0 0.5


1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4 linear geometryP-D Geometry

Load

Fac

tor l



at the ends and an elastic core is an excellent compromise between concentrated and distributed inelas-ticity elements for modeling the inelastic response of columns. The distributed inelasticity elements areparticularly suitable for the representation of the inelastic response of girders with significant influenceof gravity loads.

Figure 6.26a to c shows the response of the same frame under linear geometry with elements basedon the displacement formulation. With only one element per member, this model overestimates thecollapse load factor by almost 50%, as shown in Figure 6.26a. The cause of this discrepancy is apparentin Figures 6.26b and c, which show the distribution of section deformations and section forces atmaximum displacement, respectively. In the basic displacement formulation a constant axial strain andlinear curvature distribution is assumed, as shown in Figure 6.26b. The corresponding axial force andbending moment at each section need to satisfy the material response, while equilibrium is not satisfiedin a strict sense, but only for the element. This results in the rather unusual axial force and bendingmoment distributions of Figure 6.26c. To improve the accuracy of the results members with yielding

(a)

(b) (c)

FIGURE 6.26 (a) Load factor–top story displacement relation for three-story steel frame under linear geometry;distributed inelasticity element with displacement formulation and five integration points; layer section with 20layers. (b) Curvature (solid line) and axial strain (dashed line) distribution for three-story steel frame at maximumdisplacement; distributed inelasticity element with displacement formulation. (c) Moment (solid line) and axial force(dashed line) distribution for three-story steel frame at maximum displacement; distributed inelasticity element withdisplacement formulation.

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Load

Fac

tor l




should be subdivided into several smaller elements, thus increasing significantly the computational cost.Alternatively, higher order elements with internal nodes can be used permitting higher order polynomialsfor the displacement interpolation functions. Neither approach, however, is completely successful, par-ticularly under cyclic loading conditions, and the force formulation is preferable for inelastic frameelements with distributed inelasticity.

6.11 Dynamic Analysis

The equations of motion at the structural DOFs according to 6.2 are

(6.83)

where a dot denotes differentiation with respect to time and we have explicitly noted the variation ofapplied forces with time. M is the mass matrix of the structure, is the total acceleration in afixed reference frame, and the resisting forces in general depend on the displacement and velocity atthe global DOFs. If the resisting forces are simply linear functions of velocity and displacement, we cansimplify 6.83 to the following

(6.84)

where is the viscous damping matrix for the free DOFs of the model.

6.11.1 Free Vibration

Setting the forcing function equal to zero in 6.84 and assuming that the viscous damping is zero andthat the fixed reference frame is the base of the structure, hence , give the free vibration problem:

(6.85)

The solution of 6.85 can be expressed in terms of the vibration mode shapes ffff and natural vibrationfrequencies , as defined by the eigenvalue problem:

Kffff = w2Mffff

The number of pairs of wi,ffffi that satisfies the eigenvalue problem is equal to the number of free DOFs,although in practice much fewer modes are necessary to represent the dynamic response under earthquakeexcitation. The eigenfrequencies can be collected in a diagonal matrix and the eigenmodes in a matrixof the form

These satisfy the equation

(6.86)

The key characteristic of the mode shapes is that they are orthogonal with respect to the mass and stiffnessmatrices. Premultiplying both sides of 6.86 by gives the following relationship:

P P U U M U( ) , ˙ ˙t t- ( ) =r

P( )t ˙U t

Pr

P CU K U M Ut t( ) - - =˙ ˙

C

P( )t˙ ˙U Ut =

M U K U 0˙ + =

w

WW FF=

È

Î

ÍÍÍÍ

˘

˚

˙˙˙˙

= [ ]w

w

1

1

0 0

0 0

0 0

O L

n

nff ff

K MFF FFWW= 2

FFT



(6.87)

both sides of which are diagonal matrices because of orthogonality (Chopra, 2001). Since the modes canbe scaled arbitrarily we select a scaling such that , where I is the identity matrix. This isknown as orthonormality property of the vibration modes. With this scaling of the eigenmodes we obtainfrom 6.87 that

(6.88)

6.11.2 Modal Analysis for Linear Response

Returning to the solution of the free vibration problem in 6.85 with initial conditions on the displacement,, and velocity, , the displacement vector can be represented as summation of contributions of the

vibration modes:

(6.89)

in which Y=Y(t) are called the generalized coordinates. Usually much fewer generalized coordinates areneeded compared with the number of free DOFs. After substituting 6.89 into 6.85 and premultiplyingthe equation by we obtain

which on account of 6.88 and the orthogonality of the mode shapes gives

(6.90)

6.90 are uncoupled second order initial value problems with the following solution for mode k:

Using 6.89 the initial values for the generalized coordinates can be expressed in terms of the initialconditions by premultiplying both sides of the equation with :

For the more interesting forced vibration case with damping in 6.84 we obtain

which on account of the orthogonality properties of the vibration modes gives

(6.91)

For a general damping matrix all modes are coupled through the damping terms in 6.91. However, sincedamping is generally assumed, it is reasonable to use the so-called Rayleigh damping and express the

FF FF FF FFWWT TK M= 2

FF FFTM I=

WW FF FF2 = TK

U 0 U 0

U Y= FF

FFT

FF FF FF FFT TM Y K Y˙ + = 0

˙Y Y+ =WW2 0

Y YY

k k kk

kkt t t( ) cos

˙sin= +0

0ww

w

FFTM

FF FF FF FF

FF FF FF FF

T0

T T0

T0

T T0

MU M Y Y Y MU

MU M Y Y Y MU

= = fi =

= = fi =

0 0 0

0 0 0˙ ˙ ˙ ˙ ˙

FF FF FF FF FF FF FFT T T TM Y C Y K Y P˙ ˙ ( )+ + = t

˙ [ ] ˙ ( )Y Y Y P+ + =FF FF WW FFTC t2 T



damping matrix in terms of the mass and stiffness matrix: . Since the mode shapes areorthogonal to M, and K, they are also orthogonal to this specific form of the damping matrix. Substitutionof Rayleigh damping into (6.91) gives

(6.92)

The Rayleigh damping coefficients and are selected to match the desired damping ratio for twomodes, oftentimes the two lowest, but not always. Calling these modes k and m we can write

(6.93)

With given damping ratios and we can solve the two equations in 6.93 for and (Chopra,2001). The damping ratio for another mode is given by

6.11.3 Earthquake Excitation

In the case of earthquake excitation the support DOFs are assumed to move together through a specifiedground acceleration history, , in the global coordinate system, which generally has two componentsfor 2d problems (a horizontal and a vertical acceleration) and three components for 3d problems (twohorizontal accelerations and one vertical acceleration). The key step is to define the acceleration withrespect to the fixed reference frame as the sum of the acceleration of the support DOFs and the additionalacceleration of the free DOFs relative to the supports, expressed as follows:

(6.94)

The number of columns in R is equal to the number of specified support acceleration components. Foreach component, the column of R corresponds to the displacements of the free DOFs due to a unitdisplacement of the corresponding support. If all supports move as a rigid body, then R represents therigid-body displacement of the entire structure. For linear systems, this procedure can be generalized toinclude different motions at the supports, in which case R represents the displacements of the free DOFsdue to unit displacement of each support and is obtained by solving the static support displacementproblem (Clough and Penzien, 1993).

Under the assumption of no applied nodal loads, the substitution of 6.94 into 6.83 or 6.84 gives theequations of motion due to earthquake excitation for the nonlinear and linear models, respectively:

(6.95)

6.11.4 Numerical Integration of Equations of Motion for Linear Response

Because of the difficulty of solving the linear differential equation for arbitrary variation of forcingfunctions as a function of time (earthquake excitation is a particularly complex case in point), it isnecessary to use a numerical method to solve 6.95. In the case of linear elastic response of the structural

C M K= +a a0 1

˙ ( ) ˙ ( )Y Y Y P+ + + =a a0 12 2I W W FT t

a 0 a1

a a w z w

a a w z w

0 12

0 12

2

2

+ =

+ =

k k k

m m m

zk zm a 0 a1

zaw

a wnn

n= +Ê

ËÁˆ

¯1

20

1

˙ ( )Ug t

˙ ˙ ˙U U RUt t= + ( )g

MU P U U MRU

MU CU KU MRU

˙ , ˙ ˙

˙ ˙ ˙

+ ( ) = - ( )+ + = - ( )

r g

g

t

t



system we use modal analysis and thus integrate numerically m single DOF differential equations of theform

where . In general, the number of modes should be selected based on the frequencycontent and spatial participation of the modes as indicated by the participating mass (Chopra, 2001).

In the numerical solution of differential equations the acceleration, velocity and displacement at time are defined in terms of the acceleration, velocity and displacement at time t. To keep the notation

short we use subscript i+1 for time and subscript i for time t, respectively. Because the equationsare the same whether we are dealing with a single DOF or a multi-DOF system we use the latter in thefollowing presentation for generality.

Newmark introduced one of the most widely used methods of numerical integration in earthquakeengineering (Newmark, 1959). It uses the following relations between displacement, velocity and accel-eration at time steps i and i+1:

(6.96)

where subscript f for the free DOFs has been dropped for convenience. With the second equation in 6.96, can be expressed in terms of the displacement at time step i + 1 and the response at the previous

time step:

(6.97)

Substituting 6.97 into the first equation in 6.96 gives the velocity, :

(6.98)

We introduce now the following constants for a given time step :

and rewrite 6.97 and 6.98 in a more compact form:

(6.99)

(6.100)

˙ ˙Y Y Y Pk k k k k k t+ +( ) + = ( )a a w w0 12 2

P MR Uk t t( ) = -( ) ( )FFTg

˙

t t+ Dt t+ D

˙ ˙ ˙ ˙

˙ ˙ ˙

U U U U

U U U U U

i i i i

i i i i i

t t

t t t

+ +

+ +

= + -( ) +

= + + -ÊËÁ

ˆ¯

+

1 1

12 2

1

1

1

2

g g

b b

D D

D D D

˙U i+1

˙ ˙ ˙U U U U Ui i i i it t+ += -( ) - - -ÊËÁ

ˆ¯1 2 1

1 1 1

21

b b bD D

U i+1

˙ ˙ ˙ ˙ ˙

˙ ˙

U U U U U U U

U U U U

i i i i i i i

i i i i

tt

t

tt

+ +

+

= + -( ) + -( ) - - -ÊËÁ

ˆ¯

= -( ) + -ÊËÁ

ˆ¯

+ -ÊËÁ

ˆ¯

1 1

1

11

21

1 12

g gb

gb b

g

gb

gb

gb

DD

D

DD

Dt

Ct

Ct

Ct

C C C t0 2 1 2 3 4 5

1 1 1

21 1

21= = = = -

ÊËÁ

ˆ¯

= -ÊËÁ

ˆ¯

= -ÊËÁ

ˆ¯b b

gb b

gb

gbD D D

D

˙ ˙ ˙U U U U Ui i i i iC C C+ += -( ) - -1 0 1 1 3

˙ ˙ ˙˙U U U U Ui i i i iC C C+ += -( ) - -1 2 1 4 5



We use 6.97 and 6.98 in two ways: first we substitute the velocity and acceleration in the equations ofmotion 6.84 at time , i.e.,

(6.101)

and obtain a system of equations for the unknown displacement at time :

After collecting terms for the unknown displacement at time we get

or, in short, a system of linear equations:

(6.102)

Solving for the displacements at time from 6.102, 6.97 and 6.98 gives the velocities and acceler-ations at the free DOFs at time , thus advancing the solution of the equation of motion by onetime increment. Repeating this process for the necessary number of time steps gives the solution as afunction of time.

Because a solution of simultaneous equations is required in 6.102, Newmark’s numerical solutionbelongs to the class of implicit methods. The numerical stability and accuracy of such methods arediscussed elsewhere (Hughes, 2000).

6.11.5 Numerical Integration of Equations of Motion for Nonlinear Response

Newmark’s time integration algorithm can now be used to solve the equations of motion for a generalnonlinear model of a structure. In this case 6.101 is written as follows for time

Assuming that the resisting forces are linearly dependent on the velocity, the equations of motion become

(6.103)

Since 6.103 is a nonlinear system of equations, the Newton–Raphson algorithm is needed to solve it. Inanalogy with the static case, it is necessary to obtain the derivative of 6.103 with respect to the unknowndisplacements at time . The velocities and accelerations are expressed in terms of these displace-ments using the result from Newmark’s method. Thus, the chain rule of differentiation on 6.103 givesthe effective stiffness matrix at time step i+1:

(6.104)

t t+ D

P CU K U MUi i i i+ + + +- - =1 1 1 1˙ ˙˙

t t+ D

P C U U U U K U M U U U Ui i i i i i i i i iC C C C C C+ + + +- -( ) - -[ ] - = -( ) - -[ ]1 2 1 4 5 1 0 1 1 3 ˙ ˙ ˙ ˙

t t+ D

C C C C C C C Ci i i i i i i0 2 1 1 0 2 1 3 4 5M C K U P U M C U U M U U C+ +( ) = + +( ) + +( ) + +( )+ +˙ ˙˙ ˙ ˙˙

K U Peff effi+ =1

t t+ Dt t+ D

t t+ D

P P U U MÜi i i i+ + + +- ( ) =1 1 1 1r˙ ,

P CU P U MÜi i i i+ + + +- - ( ) =1 1 1 1˙

r

t t+ D

KU

MÜ CU P UU

P

KU

MUU

U UCU

U

U

P U

U

eff r

effr

= ∂∂

+ + ( )[ ] - ∂∂

= ∂∂ ( ) ∂∂ + ∂

∂ ( ) ∂∂ +∂ ( )∂

++ + +

++

++

+

+ ++

+

+

+

+

ii i i

ii

ii

i

i ii

i

i

i

i

11 1 1

11

11

1

1 11

1

1

1

1

˙

˙˙

˙

˙˙

˙



where we note that the applied forces do not depend on the displacements . After substituting

6.99 and 6.100 in 6.104, we conclude that the effective stiffness is similar to the linear case except for the

fact that the tangent stiffness matrix is used,

where in accordance with the tangent stiffness definition in 6.72. The force unbalance

vector of the equation in 6.103 is also needed. It expresses the amount of equilibrium error under inclusionof the mass and damping terms. It is given by

(6.105)

Substituting the expressions in 6.99 and 6.100 for the acceleration and velocity at time in a slightlymodified form in 6.105 gives the unbalanced force vector as

(6.106)

where . Note that during Newton–Raphson iterations is updated during eachiteration, while Ui, of course, remains constant and equal to the displacement values at the previouslyconverged time step. With this in mind we collect terms in 6.106 as follows

(6.107)

The first half of the right-hand side in 6.107, i.e., does not changeduring a time step and can be considered the effective applied force. The second half, namely

needs to be updated with every new estimate of the displacements during equilibrium iterations and can be regarded as the effective resisting force vector.

6.12 Applications of Linear and Nonlinear Dynamic Analysis

As an example of the structural analysis methods presented in this chapter, this section presents thenonlinear dynamic analysis of a 20-story moment-resisting steel frame building. The example building,designed for the seismic hazard in Los Angeles, has been used in the SAC studies to assess the performanceof steel moment-resisting frame buildings (Gupta and Krawinkler, 1999). The building has 20 storiesabove ground level and two basement levels. The total height above the ground is 265 ft and the storyheight is 13 ft except for the ground level story of 18 ft height. The North-South frames consist of fivebays with perimeter box columns of 15 ¥ 15 with various thicknesses and interior columns with wideflange sections varying from W24 ¥ 335 to W24 ¥ 84. The beams consist of various wide flange sectionmembers ranging between W30 ¥ 108 and W21 ¥ 50.

The structural model of the NS frame, shown in Figure 6.27, consists of two-node frame elementsconnected at nodes representing the joints. Centerline dimensions are used and the joints are assumedto be rigid. The base of the columns is hinged and the perimeter basement columns are constrained inthe horizontal direction at the ground level to represent the embedment of the basement, although amore refined model could include soil–foundation–structure interaction effects. The frame elementsrepresent distributed inelasticity with five control sections. Each element has section properties with adiscretization of typically 60 layers (fibers), although as described in Section 6.10 a smaller number can

Pi+1 Ui+1

K M C Keff t= + ++

C Ci0 2 1

KP U

Ut

r

i

i

i+

=∂ ( )∂

+

+1

1

1

P P P U MÜ Cu r= - ( ) - -+ + + +i i i i1 1 1 1U

t t+ D

P P P U M U U Ü C U U Üu r= - ( ) - - -( ) - - -( )+ + + +i i i i i i i iC C C C C C1 1 0 1 1 3 2 1 4 5D D˙ ˙

DU U Ui i i+ += -1 1 Ui+1

P P M U Ü C U Ü P U M U C Uu r= + +( ) + +( ) - ( ) - -+ + + +i i i i i i i iC C C C C C1 1 3 4 5 1 0 1 2 1˙ ˙ D D

P M U Ü C U Üi i i i iC C C C+ + +( ) + +( )1 1 3 4 5˙ ˙

- ( ) - -+ + +P U M U C Ur i i iC C1 0 1 2 1D D Ui+1



suffice. The material model for the steel is a bilinear plasticity model with 2% strain hardening ratio.The structural model has 585 free DOFs for the translational and rotational components at the nodes.

The mass of the building is represented by lumped masses at the nodes of the model. The gravityresisting frames in the building are not included in the model, but they contribute substantial P-D effectsto the moment-resisting frame. To account for the destabilizing effect of the gravity loads on the gravityresisting frames, the loads are collected to an additional column member that is attached to themoment-resisting frame by truss members. This is commonly known as leaning column approach. Thegeometric compatibility transformation for the beams and columns, including the leaning column, usesthe P-D transformation presented in Section 6.9.3.

The lower vibration mode shapes and periods, using the stiffness matrix of the building under linearelastic behavior, are shown in Figure 6.28. For dynamic analysis, Rayleigh damping is assumed based ona damping ratio of 0.02 in the first two vibration modes.

The horizontal ground motion record used in this example analysis is obtained from the simulationof a fault rupture and resulting wave propagation in a 10 km ¥ 10 km region (Bao et al., 1996). Thelocation of the station is in the forward rupture directivity region and is about 1 km from the surface

FIGURE 6.27 Model for 20-story moment-resisting frame (Gupta and Krawinkler, 1999) and ground motion forsimulations.

time (sec)

(a) Structural Model

(b) Simulated Horizontal GroundAcceleration at Base

acce

lera

tion

(g)

20

0 2 4 6 82



projection of the fault. The simulated ground motion has a pulse with large peak ground accelerationof 2 g. Although such large peak ground accelerations have not been recorded to date under this condition,the simulated record has the large pulse that is characteristic of near-source ground motion and it maybe considered a very severe case for the expected ground motion. The purpose of using the large simulatedrecord is to investigate the inelastic behavior of the frame in an extreme event. The ground motionacceleration record is shown in Figure 6.27b, and it is applied as horizontal free-field acceleration at thebase of the model. For the nonlinear dynamic analysis, the Newmark time integration method is usedwith 1800 time steps of Dt = 0.005 sec. The equilibrium equations at each time step are solved using theNewton–Raphson algorithm, and typically three or four iterations were required for convergence. On adesktop computer the analysis took three to five minutes, demonstrating the computational efficiency ofthe nonlinear dynamic analysis methods.

The history of displacement at five floors is shown in Figure 6.29. The propagation of the groundmotion pulse over the height of the building can be clearly seen. The maximum horizontal roof displace-ment of 38 in. results in residual deformation at the end of the 9-sec response history. The envelopes ofmaximum horizontal floor displacement and drift are shown in Figure 6.30. The largest story drift occursin the first floor, with a drift ratio of 0.026, and the upper floors.

FIGURE 6.28 Lower three vibration modes and periods of 20-story building.

FIGURE 6.29 Displacement history (in inches) at five floor levels in 20-story building due to horizontal groundacceleration.

(a) Mode 1: 3.9 sec (b) Mode 2: 1.4 sec (c) Mode 3: 0.8 sec

−40−20

020

−40−20

020

−40−20

020

−40−20

0

floor

1flo

or 5

floor

10

floor

15

floor

20

20

0 1 2 3 4 5 6 7 8 9−40−20

020

time (sec)



The location and maximum values of the plastic hinge rotations in the beams and columns are shownin Figure 6.31. Most of the hinges form at the end of the beams, with the largest rotation, reaching 0.030rad, in floors 3 to 5, and somewhat smaller plastic hinging in the upper four floors. Plastic hinges format the bases of the tall columns at the ground level, as would be expected in a sway mechanism, butlimited column hinging of approximately 0.01 rad occurs at perimeter and interior columns also in floors3 to 5. In addition there is limited plastic hinging in upper floor columns (16 to 18). The deformed shapein Figure 6.31 shows the residual displacements of the frame after the earthquake ends, with the shapemagnified by a scale factor of 20 for plotting.

The modeling and computation for this example were done with OpenSees — Open System forEarthquake Engineering Simulation (http://opensees.berkeley.edu). OpenSees is an open-source software

FIGURE 6.30 Envelope of maximum floor displacement and story drift for 20-story building due to horizontalground motion.

FIGURE 6.31 Maximum plastic hinge rotations in 20-story building due to horizontal ground motion. Deformedshape is the residual deformation after the earthquake (magnified by 20).

0 20 40b2b1

0123456789

1011121314151617181920

(a) displacement (in.)

floor

0 2 4

(b) story drift (in.)

0 1 2 3

(c) drift ratio (%)

0.030.020.01


http://opensees.berkeley.edu

http://opensees.berkeley.edu


framework for simulation of structural and geotechnical systems. The software design includes all themodels, element formulations and solution methods presented in this chapter, in addition to moreadvanced features. Section 6.14 provides additional information about OpenSees.

6.13 Conclusions

The objective of this chapter has been to present the methods of structural analysis in a manner thatunifies static and dynamic analysis for linear and nonlinear models. The emphasis has been on providinga consistent approach for satisfying the equations of equilibrium, compatibility and force–deformation.The methods presented in this chapter encompass the major structural analysis procedures used inearthquake-resistant design, and they recognize the increasing importance of nonlinear analysis proce-dures. The presentation has been limited to frame elements, although the methods can be extended toinclude joints, walls, diaphragms and foundation components.

Plastic analysis methods are very useful in the capacity design of structures but are limited to elastic-perfectly plastic behavior. Concentrated plasticity beam models are computationally simple and arecapable of accounting for the effects of axial force-shear-bending moment interaction. These models,however, require calibration under idealized assumptions about either the force or the deformationdistribution within the member. Furthermore, they require that the location of inelastic deformationsbe specified a priori and, typically, do not include the effect of distributed element loading. Distributedinelasticity frame elements do not suffer from the limitations of calibration and a priori specification ofthe location of inelastic deformation, but are computationally more demanding. In the displacementformulation the assumed displacement interpolation functions are not a satisfactory approximation ofthe deformation distribution in the member, unless the latter is subdivided into several elements. Adaptivemesh refinement methods have been proposed for the purpose. The force formulation offers the advan-tage that the force-interpolation functions are exact under the assumption of linear geometry. Conse-quently, a single element with several integration points (control sections) suffices for the representationof the inelastic behavior of the member. Moreover, the force formulation accounts directly for the effectof distributed element loads in girders, which can cause inelastic deformations to arise within the memberspan, instead of the member ends. Four integration points are recommended for the typical case withoutelement loads, while five integration points should be used in the presence of element loading. Byintegrating the material response over the control sections with the so-called layer or fiber section models,it is possible to directly account for the interaction of axial force and bending moment. Simple uniaxialnormal stress–strain models suffice for the purpose. Studies show that a few layers or fibers suffice toyield an excellent representation of the hysteretic response of the section. Under uniaxial bending eightto ten layers are usually sufficient for rectangular sections. For wide flange sections three layers in eachflange and four layers in the web are recommended. Under biaxial loading 25 (5 ¥ 5) to 64 (8 ¥ 8) fibersyield excellent accuracy. In a reinforced concrete section the hysteretic response is dominated by thebehavior of reinforcing steel. Thus, it is important to represent the area and distribution of reinforcementrelatively well and then use 16 (4 ¥ 4) or 25 (5 ¥ 5) fibers for the concrete. A larger number of fibersmay be necessary for distinguishing between cover concrete and core concrete confined by transversereinforcement.

Under nonlinear geometry the most significant contribution arises from the rigid-body displacementsof the frame element. It is possible to isolate this effect with the corotational formulation, in which theelement response is defined in the basic system without rigid-body modes. The end forces of the basicsystem are then transformed exactly to the local coordinate system of the undeformed element. In thiscase the tangent stiffness matrix of the element is made up of two contributions: the transformation ofthe material stiffness from the basic to the local system and the geometric stiffness matrix. Consistentapproximation of the displacement terms in the equilibrium equations and in the deformation–displace-ment relations leads to approximate theories of nonlinear geometry, such as the P-D geometric stiffness.The advantage of the presented approach is that one element type can accommodate several nonlineargeometric transformations. The effect of nonlinear geometry should be included in the nonlinear static



(pushover) analysis of buildings for average relative story drifts in excess of 1%. When the relative storydrift varies considerably over the height of the structure, it is important to include the effect of nonlineargeometry when the maximum inter-story drift exceeds 2%.

The strength softening response of structural systems under nonlinear material and geometry can betraced with load control strategies. Among these the load control strategy under constant displacementat a particular degree of freedom proves very useful in the nonlinear static (pushover) analysis of buildings,when the horizontal translation at a floor is representative of the response of the entire building, or, asingle soft story collapse mechanism forms. In more complex cases the load control strategy underconstant external work is an excellent alternative.

Most structural analysis is performed using computer software that implements one or more of theanalysis methods described in this chapter. When using computer software for analysis, the engineermust confirm that the assumptions and limitations of the models are appropriate for the structuralanalysis problem under consideration.

6.14 Future Challenges

Nonlinear structural analysis is becoming more important in earthquake-resistant design, particularlywith the development of performance-based earthquake engineering, which requires more detailedinformation about the displacements, drifts and inelastic deformation of a structure than traditionaldesign procedures. Nevertheless, many challenges remain in the field of structural analysis to meet thegoal of providing predictive simulations of the performance of a structure under earthquake excitation.The challenges encompass the needs for research in analysis and simulation, improved technology forstructural analysis software and education of students and design professionals in structural analysisadvances.

The Pacific Earthquake Engineering Research Center has undertaken the development of the OpenSystem for Earthquake Engineering Simulation (OpenSees) to address these challenges. The OpenSeessoftware is called a framework because it is an integrated set of software components used to buildsimulation applications for structural and geotechnical engineering problems. OpenSees is not a “code,”by the usual definition of a program, to solve a specific class of problems. Rather it involves a set ofclasses and objects that represent models, perform computations for solving the governing equationsand provide access to databases for the processing of results. At its most fundamental level, OpenSeescan be viewed as a set of objects that are accessed through a defined application program interface (API).The framework was designed using object-oriented principles, and is implemented in C++, a widelyused object-oriented programming language. The development of OpenSees is open-source, meaningthat all versions of the program, documentation, examples, are available on the website (http://opens-ees.berkeley.edu) for researchers, professionals and students interested in using and contributing to thesoftware.

PEER’s OpenSees research and development addresses three major future challenges. The first challengeis to improve the models of structural behavior of components, and particularly the representation ofdamage under cyclic loading. Although the computational methods for analysis have become moresophisticated in the past decade, the models used in many nonlinear analyses consist of very simpleelements. Simple nonlinear models, such as the lumped plasticity models described in Section 6.7.2,provide an indication of the nonlinear behavior of a structure, but they do not include several importantaspects of behavior that can have an appreciable effect on performance. For example, the interactionbetween flexure and shear, particularly in reinforced concrete members, is poorly understood and currentmodels only approximately attempt to capture this phenomenon, if it is included at all. Beyond thecomponent models, system models of structures are generally very approximate. Quite often these modelsare two dimensional with the approximation of three-dimensional effects. In particular, models for slabsand diaphragms are rarely used in earthquake analysis, and structural walls are represented with beam-column elements. The simple system models can only provide an approximate assessment of the failuresequence, particularly for structures with components of limited ductility and brittle behavior, which


http://opens-ees

http://opens-ees


then requires significant judgment and interpretation on the part of the engineer about the performanceof the system. Another important system aspect rarely considered in an analysis is the interaction betweenthe structure, foundation components and soil during an earthquake. In many cases, soil–founda-tion–structure interaction can affect the response and it should be included in the model and analysisof the system. As with all models, there are great challenges in validating the models using experimentaland field-observation data and characterizing the sensitivity of the modeled response in terms of theuncertainty in identifying the parameters of the models. Each of these issues has been a subject of researchin PEER and new models and approaches, particularly for soil–structure–foundation interaction, havebeen incorporated into OpenSees.

A second major area of challenge is the observation that the improvements in structural analysismethods and software have not kept pace with the rapid improvement in computing over the past decade.It is common for an engineer to perform a nonlinear analysis of a structure on a desktop computer today.However, it is often with software that uses simple models because the rate of innovation in the softwarehas not been as rapid as the hardware technology that produced the high-performance computer on thedesktop. There are tremendous opportunities with new technology for major improvements in structuralanalysis for earthquake engineering. Considering hardware, there will be increasing computational poweron not only individual computers, but also on networks of computers connected together in a designoffice or remote computational centers that will allow for parallel computation transparent to the user.This computational power will allow routine analysis of complete three-dimensional models. Perhapseven more important are the challenges that must be met to develop the analysis software of the future.New advances in software engineering of modularity and open standards hold promise in the earthquakeengineering field for advances in software development to support analysis and design applications. Inaddition to modeling and computational aspects, modular and open software can provide improvedfacilities for the visualization of structural behavior, linkages to databases for experimental data andvalidation studies and design databases. Software can provide support for engineers to collaborate, notonly on analysis, but also on integrating the analysis with the design process. The OpenSees frameworkaddresses these shortcomings by providing well-defined interfaces to software components for modelingand analysis, and also software tools for equation solvers, visualization, databases and distributed networkcommunication.

The third challenge is educating future and current earthquake engineers in modern methods ofstructural analysis and application to earthquake-resistant design, and also implementation in moderncomputational environments, such as OpenSees. This chapter has presented the fundamentals of analysisin a way that can be integrated into undergraduate and graduate curricula, serve as a framework forfuture advances and provide the necessary background for engineers to use nonlinear analysis methodswith confidence. It is hoped that the consistent exposition of the fundamentals and examples of structuralanalysis applications is a step toward the goal of improving the education of engineers on this importantsubject.

References

Argyris, J. H., Hilpert, O., Malejannakis, G. A. and Scharpf, D. W. (1979). On the Geometrical Stiffnessof a Beam in Space: a Consistent Virtual Work Approach. Computer Methods in Applied Mechanicsand Engineering, 20, 105–131.

ATC. (1996a). Improved Seismic Design Criteria for California Bridges: Provisional Recommendations.ATC-32, Applied Technology Council.

ATC. (1996b). Seismic Evaluation and Retrofit of Concrete Buildings. ATC-40, Applied TechnologyCouncil.

Bao, H., Bielak, J., Ghattas, O., O'Hallaron, D. R., Kallivokas, L. F., Shewchuk, J. R. and Xu, J. (1996).Earthquake Ground Motion Modeling on Parallel Computers. Supercomputing '96, Pittsburgh,Pennsylvania.

Bathe, K.-J. (1995). Finite Element Procedures, Prentice Hall, New York.



Bergan, P. G. (1978). Solution Techniques for Nonlinear Finite Element Problems. International Journalof Numerical Methods and Engineering, 12, 1677–1696.

CEB, C. E.-I. d. B. (1996). RC elements under cyclic loading. 230, CEB, Comité Euro-International duBéton.

Chopra, A. K. (2001). Dynamics of Structures: Theory and Applications to Earthquake Engineering, 2nd.ed., Prentice Hall, New York, N.Y.

Ciampi, V. and Carlesimo, L. (1986). A Nonlinear Beam Element for Seismic Analysis of Structures. 8thEuropean Conference on Earthquake Engineering, Lisbon, 6.3/73–6.3/80.

Clarke, M. J. and Hancock, G. J. (1990). A study of incremental-iterative strategies for non-linear analyses.International. Journal of Numerical Methods and Engineering, 29, 1365–1391.

Clough, R. W., Benuska, K. L. and Wilson, E. L. (1965). Inelastic Earthquake Response of Tall Buildings.Third World Conference on Earthquake Engineering, Wellington, New Zealand.

Clough, R. W. and Penzien, J. (1993). Dynamics of Structures, 2nd ed., Prentice Hall, New York.Crisfield, M. A. (1990). A consistent co-rotational formulation for non-linear, three dimensional, beam

elements. Computer Methods in Applied Mechanics and Engineering, 81, 131–150.Crisfield, M. A. (1991). Non-Linear Finite Element Analysis of Solids and Structures, John Wiley & Sons,

West Sussex.Elias, Z. M. (1986). Theory and Methods of Structural Analysis, John Wiley & Sons, New York.FEMA (1997). NEHRP Guidelines for the Seismic Rehabilitation of Buildings. FEMA 273, Federal Emer-

gency Management Agency, Washington, D.C.FEMA (2000a). Prestandard and Commentary for the Seismic Rehabilitation of Buildings. FEMA 356,

Federal Emergency Management Agency, Washington, D.C.FEMA (2000b). Recommended Seismic Design Criteria for New Steel Moment-Frame Buildings. FEMA

350, Federal Emergency Management Agency, Washington, D.C.FEMA (2000c). Recommended Seismic Evaluation and Upgrade Criteria for Existing Welded Steel

Moment-Frame Buildings. FEMA 351, Federal Emergency Management Agency, Washington, D.C.FEMA (2001). NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other

Structures, 2000 Edition: Part 1-Provisions (FEMA 368), Part 2-Commentary (FEMA 369). FederalEmergency Management Agency, Washington, D.C.

Filippou, F. C., Popov, E. P. and Bertero, V. V. (1983). Effects of Bond Deterioration on Hysteretic Behaviorof Reinforced Concrete Joints. UCB/EERC-83/19, Earthquake Engineering Research Center, Uni-versity of California, Berkeley.

Giberson, M. F. (1967). The Response of Nonlinear Multistory Structures Subjected to EarthquakeExcitation, Ph.D. Dissertation, California Institute of Technology, Pasadena.

Gupta, A. and Krawinkler, H. (1999). Seismic demands for performance evaluation of steel momentresisting frame structures. Report No. 132, John A. Blume Earthquake Engineering Center, StanfordUniversity.

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Structural Division, ASCE, 95, 2543–2563.Livesley, R.K. (1975). Matrix Methods of Structural Analysis, 2nd. ed., Pergamon Press, Oxford, U.K.McGuire, W., Gallagher, R. H. and Ziemian, R. D. (2000). Matrix Structural Analysis, John Wiley & Sons,

New York.Menegotto, M. and Pinto, P. E. (1973). Method of Analysis for Cyclically Loaded Reinforced Concrete

Plane Frames Including Changes in Geometry and Non-Elastic Behavior of Elements under Com-bined Normal Force and Bending. IABSE Symposium on Resistance and Ultimate Deformability ofStructures Acted on by Well Defined Repeated Loads, Lisbon, 15–22.

Neuenhofer, A. and Filippou, F. C. (1997). Evaluation of nonlinear frame finite element models. Journalof Structural Engineering, ASCE, 123, 958–966.



Newmark, N. M. (1959). A method of computation for structural dynamics. Journal of EngineeringMechanics Division, ASCE, 85(EM3), 69–74.

Paulay, T. and Priestley, M. J. N. (1992). Seismic Design of Reinforced Concrete and Masonry Buildings,Wiley Interscience, New York.

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UBC (1997). Uniform Building Code, International Council of Building Code Officials.

Glossary

basic element forces — set of independent element forces in equilibrium equations of element free bodyconcentrated or lumped inelasticity — inelastic deformations may arise at specific locations along the

element axis, typically at the element endscorotational formulation — element force-–deformation relation is set up in a reference system that

moves with the element as it deformsdistributed inelasticity — inelastic deformations may arise anywhere along element axiselement deformations — relative element end displacements excluding rigid body modeslayer or fiber section — integration of material response over the cross section in one or two dimensions

by midpoint ruleload factor control — relations for load factor adjustment during load incrementation and/or equilib-

rium iterationslocal coordinate system — orthogonal Cartesian coordinate system with x-axis coinciding with the line

connecting the end nodes of the elementmodal analysis — decomposition of linear dynamic response in eigenvector contributionsnonlinear geometry — large displacement compatibility relations and equilibrium in the deformed

configurationnonlinear response — nonlinear relation between displacements at global degrees of freedom and cor-

responding resisting forcesP-DDDD geometric stiffness — small displacement compatibility relations and equilibrium in the deformed

configuration for axial force effect onlypush-over analysis — step-by-step nonlinear analysis to collapse under constant gravity loads and a

reference lateral force vector with gradually increasing load factorsection deformations — deformation measures of infinitesimal slice of frame elementsection forces — resultant forces at section of frame elementstructural model — collection of points (structural nodes) in space interconnected by structural ele-

ments


methods of analysis for earthquake- resistant structures

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